Exhibit 5.1

Reed Smith LLP

599 Lexington Avenue
New York, NY 10022-7650

+1 212 521 5400

Fax +1 212 521 5450

reedsmith.com

September 13, 2024

Adagio Medical Holdings, Inc.

26051 Merit Circle, Suite 102

Laguna Hills, CA 92653

Ladies and Gentlemen:

We have acted as counsel to Adagio Medical Holdings, Inc. (f/k/a Aja Holdco, Inc.), a Delaware corporation (the “Company”), in connection with the preparation and filing by the Company with the Securities and Exchange Commission (the “Commission”) of a registration statement on Form S-1 (the “Registration Statement”) and the related prospectus filed with the Registration Statement (the “Prospectus”), covering the registration of up to (i) 7,951,913 shares of common stock, $0.0001 par value per share, of the Company (the “Common Stock”), issued pursuant to those certain amended and restated subscription agreements, dated July 31, 2024 (the “PIPE Shares” and the “PIPE Subscription Agreements”) between the Company, Arya Sciences Acquisition Corp IV, a Cayman islands exempted Company (“ARYA”), Perceptive Life Sciences Master Fund, Ltd (“Perceptive”), a Cayman Islands exempted company and certain other investors, (ii) up to 2,354,100 shares (the “Sponsor Shares”) issued to ARYA Sciences Holdings IV, a Cayman Islands exempted company (the “Sponsor”) pursuant to the Business Combination Agreement dated as of February 13, 2024, by and among the Company, ARYA, Adagio Medical Inc. (“Adagio Medical”), and the other parties thereto, as amended (the “Business Combination Agreement”), (iii) up to 582,475 shares, consisting of 492,475 shares of Common Stock issued to certain of the Company's affiliates (the “Affiliate Shares”) as consideration in connection with the business combination (the “Business Combination”) pursuant to the Business Combination Agreement, and 30,000 shares of Common Stock issued to each of Michael Henderson, Todd Wider and Leslie Trigg that were originally issued as Class B ordinary shares of ARYA and exchanged for shares of Common Stock pursuant to the Business Combination Agreement (the “Director Shares,” and together with the PIPE Shares, the Sponsor Shares and the Affiliate Shares, the “Issued Shares”), (iv) 7,951,913 shares of Common Stock (the “PIPE Warrant Shares”) issuable upon exercise of the warrants to purchase shares of Common Stock (the “PIPE Warrants”) issued pursuant to the Warrant Agreement, dated July 31, 2024 (including the form of warrant certificate included therein, the “Base Warrant Agreement”) and the PIPE Subscription Agreements, (v) 900,000 shares of Common Stock (the “Convert Warrant Shares”) issuable upon exercise of warrants to purchase shares of Common Stock (the “Convert Warrants”), issued pursuant to the Convert Warrant Agreement, dated July 31, 2024 (including the form of warrant certificate included therein, the “Convert Warrant Agreement”) and that certain securities purchase agreement, dated February 13, 2024, by and among the Company and certain investors named therein, and any assignment thereunder (the “Convertible Security Subscription Agreement”), and that certain note purchase agreement, dated February 13, 2024, by and among the Company, Adagio Medical and Perceptive (the “2024 Bridge Financing Note Subscription Agreement” and together with the Convertible Security Subscription Agreement, the “Convert Subscription Agreements” ), (vi) 670,000 shares of Common Stock (the “Pre-Funded Warrant Shares”) issuable upon exercise of pre-funded warrants (the “Pre-Funded Warrants”) to purchase shares of Common Stock issued pursuant to the PIPE Subscription Agreements and that certain Pre-Funded Warrant Agreement, dated July 31, 2024 (including the form of warrant certificate included therein, the “Pre-Funded Warrant Agreement”), (vii) up to 1,147,500 shares of Common Stock issued to the Sponsor pursuant to the Business Combination Agreement that are subject to share trigger price vesting (the “Trigger Price Vesting Shares”) and will vest if, prior to the tenth anniversary of the closing of the Business Combination (the “Closing”), the post-Closing share price of Common Stock equals or exceeds $24.00 per share for any 20 trading days within any 30 trading day period, (viii) up to 11,858,081 shares of Common Stock issuable upon the conversion of those certain 13% senior secured convertible notes (the “Convertible Note Shares”) issued pursuant to the Convertible Security Subscription Agreement; and (ix) 1,000 shares of Common Stock (the “Option Shares”) issuable upon exercise of certain stock

options (the “Options”) granted to the Company's Chief Operating Officer and assumed by the Company in connection with the consummation of the Business Combination pursuant to the terms of the Business Combination Agreement.

This opinion letter is being furnished in accordance with the requirements of Item 601(b)(5) of Regulation S-K under the Securities Act.

We have reviewed originals or copies of (a) the Registration Statement, (b) the Business Combination Agreement, (c) the Company’s certificate of incorporation and bylaws, each as currently in effect, (d) the PIPE Subscription Agreements, (e) Base Warrant Agreement, (f) the Pre-Funded Warrant Agreement, (g) the Convertible Security Subscription Agreement, (h) the Convert Warrant Agreement, (i) the Investor Rights Agreement, dated as of February 13, 2024, by and among the Company, ARYA, the Sponsor, Perceptive and the other parties thereto (the “Investor Rights Agreement”), and (j) the Registration Rights Agreement, dated as of July 31, 2024, by and between the Company, Perceptive Life Sciences Master Fund, Ltd. and each holder thereto (the “Convert Registration Rights Agreement” and together with the Business Combination Agreement, the PIPE Subscription Agreements, the Base Warrant Agreement, the Pre-Funded Warrant Agreement, the Convertible Security Subscription Agreement, the Convert Warrant Agreement and the Investor Rights Agreement, the “Transaction Documents”), and such other corporate records, agreements and documents of the Company, certificates or comparable documents of public officials and officers of the Company and have made such other investigations as we have deemed necessary as a basis for the opinions set forth below.

In rendering the opinion set forth below, we have assumed:

the genuineness of all signatures;

the legal capacity of natural persons;

the authenticity of all documents submitted to us as originals;

the conformity to original documents of all documents submitted to us as duplicates or conformed copies; and

as to matters of fact, the truthfulness of the representations and warranties made or to be made (as applicable) by the parties to the Transaction Documents and the truthfulness of the representations made in certificates or comparable documents of public officials and officers of the Company.

We have not independently established the validity of the foregoing assumptions.

Based upon the foregoing, and subject to the qualifications, assumptions and limitations stated herein, we are of the opinion that:

The Issued Shares have been duly authorized and are validly issued, fully paid and non-assessable.

The PIPE Warrant Shares, the Convert Warrant Shares and the Pre-Funded Warrant Shares have been duly authorized and, when issued and paid for upon exercise of the PIPE Warrants, Convert Warrants and Pre-Funded Warrants, in accordance with the terms of the Base Warrant Agreement, Convert Warrant Agreement and Pre-Funded Warrant Agreement, as applicable, will be validly issued, fully paid and nonassessable.

The Trigger Price Vesting Shares have been duly authorized and, when issued and delivered in accordance with the terms of the Business Combination Agreement, will be validly issued, fully paid and nonassessable.

The Option Shares have been duly authorized and, when issued and paid for upon exercise of the Options in accordance with the terms of such Options, will be validly issued, fully paid and nonassessable.

The Convertible Note Shares have been duly authorized and, when issued and delivered upon conversion of the Convertible Notes in accordance with the terms of the Convertible Security Subscription Agreement, will be validly issued, fully paid and nonassessable.

The opinions stated herein are subject to the qualifications that we express no opinion as to the applicability of, compliance with, or effect of (i) public policy considerations which may limit the rights of parties to obtain certain

remedies, (ii) any provision waiving the right to object to venue in any court; and (iii) any agreement to submit to the jurisdiction of any federal court.

Our opinion is limited to the General Corporation Law of the State of Delaware and the laws of the State of New York (the “Opined-on Law”) and we do not express any opinion herein concerning any other law. We advise you that issues addressed by this letter may be governed in whole or in part by other laws, but we express no opinion as to whether any relevant difference exists between the laws upon which our opinions are based and any other laws which may actually govern. This opinion letter speaks only as of its date.

We hereby consent to the filing of this opinion letter as Exhibit 5.1 to the Registration Statement and to the use of our name under the caption “Legal Matters” in the prospectus forming part of the Registration Statement and any amendments thereto. In giving such consent, we do not hereby admit that we are within the category of persons whose consent is required under Section 7 of the Securities Act, and the rules and regulations of the Commission promulgated thereunder.

	Very truly yours,

	/s/ REED SMITH LLP

	Reed Smith LLP