1. General Data Protection Regulation (EU) 2016/679.
Article 26(1): "Where two or more controllers jointly determine the purposes and means of processing, they shall be joint controllers." This directly defines the scenario described.
Article 4(7): Defines a 'controller' as the entity that "determines the purposes and means of the processing of personal data." The joint nature of this determination is key.
2. European Data Protection Board (EDPB), Guidelines 07/2020 on the concepts of controller and processor in the GDPR, Version 2.1, Adopted on 7 July 2021.
Paragraph 52: "Joint controllership exists when different entities process the same or a set of personal data for a joint purpose and have jointly defined the means for this processing."
Paragraph 53: "The joint participation can take the form of a common decision taken by two or more entities or result from converging decisions... where the decisions complement each other and are necessary for the processing to take place..." This describes the businesses' agreement to use a single system for a shared goal.
Section 2.1, Example 1 (p. 23): Discusses a research project where several institutes jointly agree on the purpose, data, and means, concluding they are joint controllers. This provides a direct parallel to the business collaboration in the question.