Exhibit 10.5

PRIVATE PLACEMENT UNITS PURCHASE AGREEMENT

THIS PRIVATE PLACEMENT UNITS PURCHASE AGREEMENT, dated as of November 6, 2025 (as it may from time to time be amended, this “Agreement”), is entered into by and between Tailwind 2.0 Acquisition Corp., a Cayman Islands exempted company (the “Company”), and Cohen & Company Capital Markets, a division of Cohen & Company Securities, LLC (the “Purchaser”).

WHEREAS, the Company intends to consummate an initial public offering of the Company’s units (the “Public Offering”), each unit consisting of one Class A Ordinary Share, par value $0.0001 per share, of the Company (an “Ordinary Share”), and one right to receive one-tenth (1/10) of one Ordinary Share upon the consummation of the Company’s initial business combination as set forth in the Company’s Registration Statement on Form S-1, filed with the U.S. Securities and Exchange Commission (the “SEC”), File No. 333-289546, under the Securities Act of 1933, as amended (the “Securities Act”). The Purchaser has agreed to purchase an aggregate of 150,000 units (or 172,500 private placement units if the over-allotment option in connection with the Public Offering is exercised in full) (the “Private Placement Units”), each Private Placement Unit consisting of one Ordinary Share and one right (the “Private Share Right”) to receive one-tenth (1/10) of one Ordinary Share upon the consummation of an initial business combination.

NOW THEREFORE, in consideration of the mutual promises contained in this Agreement and other good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, the parties to this Agreement hereby, intending legally to be bound, agree as follows:

AGREEMENT

Section 1. Authorization, Purchase and Sale; Terms of the Private Placement Units.

A. Authorization of the Private Placement Units. The Company has duly authorized the issuance and sale of the Private Placement Units (including the securities comprising such units) to the Purchaser.

B. Purchase and Sale of the Private Placement Units.

(i) On the date of the consummation of the Public Offering or on such earlier time and date as may be mutually agreed by the Purchaser and the Company (the “IPO Closing Date”), the Company shall issue and sell to the Purchaser, and the Purchaser shall purchase from the Company, an aggregate of 150,000 Private Placement Units at a price of $10.00 per unit for an aggregate purchase price of $1,500,000 (the “Purchase Price”), which shall be paid by wire transfer of immediately available funds to the Company at least one business day prior to the IPO Closing Date in accordance with the Company’s wiring instructions. On the IPO Closing Date, upon the payment by the Purchaser of the Purchase Price, the Company, at its option, shall deliver a certificate evidencing the Private Placement Units purchased by the Purchaser on such date duly registered in the Purchaser’s name to the Purchaser, or effect such delivery in book-entry form.

(ii) On the date of any closing of the over-allotment option, if any, in connection with the Public Offering or on such earlier time and date as may be mutually agreed by the Purchaser and the Company (each such date, an “Over-allotment Closing Date”, and each Over-allotment Closing Date (if any) and the IPO Closing Date, a “Closing Date”), the Company shall issue and sell to the Purchaser, and the Purchaser shall purchase from the Company, up to 22,500 Private Placement Units (or, to the extent the over-allotment option is not exercised in full, a lesser number of Private Placement Units in proportion to the portion of the over-allotment option that is exercised) at a price of $10.00 per unit for an aggregate purchase price of up to $225,000 (the “Over-allotment Purchase Price”) which shall be paid by wire transfer of immediately available funds to the Company at least one business day prior to the Over-allotment Closing Date in accordance with the Company’s wiring instructions. On the Over-allotment Closing Date, upon the payment by the Purchaser of the Over-allotment Purchase Price, the Company, at its option, shall deliver a certificate evidencing the Private Placement Units purchased on such date duly registered in the Purchaser’s name to the Purchaser, or effect such delivery in book-entry form.

C. Terms of the Private Placement Units.

(i) Each Private Placement Unit shall have the terms set forth herein. Each Private Share Right shall have the terms set forth in a Share Rights Agreement dated the date hereof (the “Share Rights Agreement”) by and between the Company and Lucky Lucko, Inc. d/b/a Efficiency (the “Share Rights Agent”).

(ii) At the time of the closing of the Public Offering, the Company and the Purchaser shall enter into a registration rights agreement (the “Registration Rights Agreement”) pursuant to which the Company will grant certain registration rights to the Purchaser relating to the Private Placement Units (and the underlying securities contained therein).

(iii) The Purchaser acknowledges and agrees that the Private Placement Units (including the securities comprising such units) and the related registration rights will be deemed compensation by the Financial Industry Regulatory Authority (“FINRA”) and will therefore, pursuant to Rule 5110(e) of the FINRA Manual, be subject to lock-up for a period of 180 days immediately following the date of effectiveness or commencement of sales in the Public Offering, subject to certain limited exceptions to permitted transferees in accordance with FINRA Rule 5110(e)(2). Additionally, the Private Placement Units (including the securities comprising such units) and the related registration rights may not be sold, transferred, assigned, pledged or hypothecated during the foregoing 180 day period following the effective date of the registration statement except to any underwriter or selected dealer participating in the Public Offering and the bona fide officers, directors or members of the Purchaser and any such participating underwriter or selected dealer. Additionally, the Private Placement Units (including the securities comprising such units) and the related registration rights will not be the subject of any hedging, short sale, derivative, put or call transaction that would result in the economic disposition of such securities by any person for a period of 180 days immediately following the date of effectiveness or commencement of sales in the Public Offering. Notwithstanding anything contained in this Agreement or the Share Rights Agreement to the contrary, so long any Private Share Rights are held by the Representative or its designees or affiliates, such Private Share Rights may not be exercised after five years from the commencement of sales in the Public Offering.

Section 2. Representations and Warranties of the Company. As a material inducement to the Purchaser to enter into this Agreement and purchase the Private Placement Units, the Company hereby represents and warrants to the Purchaser (which representations and warranties shall survive the Closing Date) that:

A. Incorporation and Corporate Power. The Company is an exempted company duly incorporated, validly existing and in good standing under the laws of the Cayman Islands and is qualified to do business in every jurisdiction in which the failure to so qualify would reasonably be expected to have a material adverse effect on the financial condition, operating results or assets of the Company. The Company possesses all requisite corporate power and authority necessary to carry out the transactions contemplated by this Agreement and the Share Rights Agreement.

B. Authorization; No Breach.

(i) The execution, delivery and performance of this Agreement and the Private Placement Units (including the securities comprising such units) have been duly authorized by the Company as of the Closing Date. This Agreement constitutes the valid and binding obligation of the Company, enforceable in accordance with its terms, subject to bankruptcy, insolvency, fraudulent conveyance, reorganization, moratorium and other laws of general applicability relating to or affecting creditors’ rights and to general equitable principles (whether considered in a proceeding in equity or law). Upon issuance in accordance with, and payment pursuant to, the terms of this Agreement, the Private Placement Units and the Private Share Rights will constitute valid and binding obligations of the Company, enforceable in accordance with their respective terms as of the Closing Date.

(ii) The execution and delivery by the Company of this Agreement, the issuance and sale of the Private Placement Units and underlying securities, and the fulfillment of, and compliance with, the respective terms hereof and thereof by the Company, do not and will not as of the Closing Date (a) conflict with or result in a breach of the terms, conditions or provisions of, (b) constitute a default under, (c) result in the creation of any lien, security interest, charge or encumbrance upon the Company’s equity or assets under, (d) result in a violation of, or (e) require any authorization, consent, approval, exemption or other action by or notice or declaration to, or filing with, any court or administrative or governmental body or agency pursuant to the Amended and Restated Memorandum and Articles of Association of the Company in effect on the date hereof or as may be amended at or prior to completion of the contemplated Public Offering, or any material law, statute, rule or regulation to which the Company is subject, or any agreement, order, judgment or decree to which the Company is subject, except for any filings required after the date hereof under federal or state securities laws.

C. Title to Securities. Upon issuance in accordance with, and payment pursuant to, the terms hereof, the Private Placement Units and underlying securities will be duly and validly issued, fully paid and nonassessable. On the date of issuance of the Private Placement Units, the Ordinary Shares issuable upon conversion of the Private Share Rights shall have been reserved for issuance by the Company. Upon issuance in accordance with, and payment pursuant to, the terms hereof, and upon registration in the books maintained by or on behalf of the Company for the registration and transfer of the Private Placement Units or the Company’s register of members (in the case of the Ordinary Shares issuable upon conversion of the Private Share Rights), the Purchaser will have good title to the Private Placement Units and underlying securities, free and clear of all liens, claims and encumbrances of any kind, other than (i) transfer restrictions hereunder and under the other agreements contemplated hereby, (ii) transfer restrictions under federal and state securities laws, and (iii) liens, claims or encumbrances imposed due to the actions of the Purchaser.

D. Governmental Consents. No permit, consent, approval or authorization of, or declaration to or filing with, any governmental authority is required in connection with the execution, delivery and performance by the Company of this Agreement or the consummation by the Company of any other transactions contemplated hereby.

E. Regulation D Qualification. Neither the Company nor, to its knowledge, any of its affiliates, members, officers, directors or beneficial shareholders of 20% or more of its outstanding securities, has experienced a disqualifying event as enumerated pursuant to Rule 506(d) of Regulation D under the Securities Act.

F. Underwriting Agreement. The representations and warranties of the Company set forth in the Underwriting Agreement (as defined below) are hereby incorporated herein.

Section 3. Representations and Warranties of the Purchaser. As a material inducement to the Company to enter into this Agreement and issue and sell the Private Placement Units to the Purchaser, the Purchaser hereby represents and warrants to the Company (which representations and warranties shall survive each Closing Date) that:

A. Organization and Requisite Authority. The Purchaser possesses all requisite power and authority necessary to carry out the transactions contemplated by this Agreement.

B. Authorization; No Breach.

(i) This Agreement constitutes a valid and binding obligation of the Purchaser, enforceable in accordance with its terms, subject to bankruptcy, insolvency, fraudulent conveyance, reorganization, moratorium and other laws of general applicability relating to or affecting creditors’ rights and to general equitable principles (whether considered in a proceeding in equity or law).

(ii) The execution and delivery by the Purchaser of this Agreement and the fulfillment of and compliance with the terms hereof by the Purchaser does not and shall not as of each Closing Date conflict with or result in a breach by the Purchaser of the terms, conditions or provisions of any agreement, instrument, order, judgment or decree to which the Purchaser is subject.

C. Investment Representations.

(i) The Purchaser is acquiring the Private Placement Units (and underlying securities) and, upon conversion of the Private Share Rights, the Ordinary Shares underlying the Private Share Rights (collectively, the “Securities”), for the Purchaser’s own account, for investment purposes only and not with a view towards, or for resale in connection with, any public sale or distribution thereof.

(ii) The Purchaser is an “accredited investor” as such term is defined in Rule 501(a)(3) of Regulation D, and the Purchaser has not experienced a disqualifying event as enumerated pursuant to Rule 506(d) of Regulation D under the Securities Act.

(iii) The Purchaser understands that the Securities are being offered and will be sold to it in reliance on specific exemptions from the registration requirements of the United States federal and state securities laws and that the Company is relying upon the truth and accuracy of, and the Purchaser’s compliance with, the representations and warranties of the Purchaser set forth herein in order to determine the availability of such exemptions and the eligibility of the Purchaser to acquire such Securities.

(iv) The Purchaser did not decide to enter into this Agreement as a result of any general solicitation or general advertising within the meaning of Rule 502(c) of Regulation D under the Securities Act.

(v) The Purchaser has been furnished with all materials relating to the business, finances and operations of the Company and materials relating to the offer and sale of the Securities which have been requested by the Purchaser. The Purchaser has been afforded the opportunity to ask questions of the executive officers and directors of the Company. The Purchaser understands that its investment in the Securities involves a high degree of risk and it has sought such accounting, legal and tax advice as it has considered necessary to make an informed investment decision with respect to the acquisition of the Securities.

(vi) The Purchaser understands that no United States federal or state agency or any other government or governmental agency has passed on or made any recommendation or endorsement of the Securities or the fairness or suitability of the investment in the Securities by the Purchaser nor have such authorities passed upon or endorsed the merits of the offering of the Securities.

(vii) The Purchaser understands that: (a) the Securities have not been and are not being registered under the Securities Act or any state securities laws, and may not be offered for sale, sold, assigned or transferred unless (1) subsequently registered thereunder or (2) sold in reliance on an exemption therefrom; and (b) except as specifically set forth in the Registration Rights Agreement, neither the Company nor any other person is under any obligation to register the Securities under the Securities Act or any state securities laws or to comply with the terms and conditions of any exemption thereunder. While the Purchaser understands that Rule 144 is not available for the resale of securities initially issued by shell companies (other than business combination related shell companies) or issuers that have been at any time previously a shell company, the Purchaser understands that Rule 144 includes an exception to this prohibition if the following conditions are met: (i) the issuer of the securities that was formerly a shell company has ceased to be a shell company; (ii) the issuer of the securities is subject to the reporting requirements of Section 13 or 15(d) of the Securities Exchange Act of 1934, as amended (the “Exchange Act”); (iii) the issuer of the securities has filed all Exchange Act reports and material required to be filed, as applicable, during the preceding 12 months (or such shorter period that the issuer was required to file such reports and materials), other than Form 8-K reports; and (iv) at least one year has elapsed from the time that the issuer filed current Form 10 type information with the SEC reflecting its status as an entity that is not a shell company.

(viii) The Purchaser has such knowledge and experience in financial and business matters, knows of the high degree of risk associated with investments in the securities of companies in the development stage such as the Company, is capable of evaluating the merits and risks of an investment in the Securities and is able to bear the economic risk of an investment in the Securities in the amount contemplated hereunder for an indefinite period of time. The Purchaser has adequate means of providing for its current financial needs and contingencies and will have no current or anticipated future needs for liquidity which would be jeopardized by the investment in the Securities. The Purchaser can afford a complete loss of its investment in the Securities.

Section 4. Conditions of the Purchaser’s Obligations. The obligation of the Purchaser to purchase and pay for the Private Placement Units is subject to the fulfillment, on or before each Closing Date, of each of the following conditions:

A. Representations and Warranties. The representations and warranties of the Company contained in Section 2 shall be true and correct at and as of such Closing Date as though then made.

B. Performance. The Company shall have performed and complied with all agreements, obligations and conditions contained in this Agreement that are required to be performed or complied with by it on or before such Closing Date.

C. No Injunction. No litigation, statute, rule, regulation, executive order, decree, ruling or injunction shall have been enacted, entered, promulgated or endorsed by or in any court or governmental authority of competent jurisdiction or any self-regulatory organization having authority over the matters contemplated hereby, which prohibits the consummation of any of the transactions contemplated by this Agreement.

D. Share Rights Agreement, Registration Rights Agreement and Underwriting Agreement. The Company shall have entered into the Share Rights Agreement with the Share Rights Agent and the Registration Rights Agreement, each on terms satisfactory to the Purchaser; the Company shall have satisfied the conditions set forth in Section 4 of the Underwriting Agreement, dated as of the date hereof, by and between the Company and the Purchaser, as representative of the underwriters named therein (the “Underwriting Agreement”).

Section 5. Conditions of the Company’s Obligations. The obligations of the Company to the Purchaser under this Agreement are subject to the fulfillment, on or before each Closing Date, of each of the following conditions:

A. Representations and Warranties. The representations and warranties of the Purchaser contained in Section 3 shall be true and correct at and as of such Closing Date as though then made.

B. Performance. The Purchaser shall have performed and complied with all agreements, obligations and conditions contained in this Agreement that are required to be performed or complied with by the Purchaser on or before such Closing Date.

C. Corporate Consents. The Company shall have obtained the consent of its Board of Directors authorizing the execution, delivery and performance of this Agreement and the Share Rights Agreement and the issuance and sale of the Private Placement Units hereunder.

D. No Injunction. No litigation, statute, rule, regulation, executive order, decree, ruling or injunction shall have been enacted, entered, promulgated or endorsed by or in any court or governmental authority of competent jurisdiction or any self-regulatory organization having authority over the matters contemplated hereby, which prohibits the consummation of any of the transactions contemplated by this Agreement.

E. Share Rights Agreement. The Company shall have entered into the Share Rights Agreement with the Share Rights Agent and the Registration Rights Agreement, each on terms satisfactory to the Company.

Section 6. Termination. This Agreement may be terminated at any time after December 31, 2025 upon the election by either the Company or the Purchaser upon written notice to the other party if the closing of the Public Offering does not occur prior to such date.

Section 7. Survival of Representations and Warranties. All of the representations and warranties contained herein shall survive each Closing Date.

Section 8. Definitions. Terms used but not otherwise defined in this Agreement shall have the meaning assigned to such terms in the registration statement on Form S-1 the Company has filed with the U.S. Securities and Exchange Commission, under the Securities Act.

Section 9. Miscellaneous.

A. Successors and Assigns. Except as otherwise expressly provided herein, all covenants and agreements contained in this Agreement by or on behalf of any of the parties hereto shall bind and inure to the benefit of the respective successors of the parties hereto whether so expressed or not. Notwithstanding the foregoing or anything to the contrary herein, the parties may not assign this Agreement, other than assignments by the Purchaser to affiliates thereof (including, without limitation one or more of its members).

B. Severability. Whenever possible, each provision of this Agreement shall be interpreted in such manner as to be effective and valid under applicable law, but if any provision of this Agreement is held to be prohibited by or invalid under applicable law, such provision shall be ineffective only to the extent of such prohibition or invalidity, without invalidating the remainder of this Agreement.

C. Counterparts. This Agreement may be executed simultaneously in two or more counterparts, none of which need contain the signatures of more than one party, but all such counterparts taken together shall constitute one and the same agreement.

D. Descriptive Headings; Interpretation. The descriptive headings of this Agreement are inserted for convenience only and do not constitute a substantive part of this Agreement. The use of the word “including” in this Agreement shall be by way of example rather than by limitation.

E. Governing Law. This Agreement shall be deemed to be a contract made under the laws of the State of New York and for all purposes shall be construed in accordance with the internal laws of the State of New York.

F. Amendments. This Agreement may not be amended, modified or waived as to any particular provision, except by a written instrument executed by all parties hereto.

[Signature Page Follows]

IN WITNESS WHEREOF, the parties hereto have executed this Agreement to be effective as of the date first set forth above.

COMPANY:

TAILWIND 2.0 ACQUISITION CORP.

By:	/s/ Sharo M. Atmeh
	Name:	Sharo M. Atmeh
	Title:	Chief Executive Officer

PURCHASER:

COHEN & COMPANY CAPITAL MARKETS, A DIVISION OF COHEN & COMPANY SECURITIES, LLC

By:	/s/ Jerry Serowik
	Name:	Jerry Serowik
	Title:	Managing Director

[Signature Page to Private Placement Units Purchase Agreement]

Tailwind 2.0 Acquisition Corp. 8-K