Your pay stub says the University of Michigan.  Michigan Medicine is a wholly-owned subsidiary of the University.  The HOA is the sole and exclusive bargaining agent recognized by the University.

None of the affiliated health systems or participating sites where you are required to rotate are joint employers.  While the VA wants to treat you as if you were their employee, you are not.  In fact, SJMHS is subject to the National Labor Relations Act.  They are essentially a national religious organization.  All University of Michigan employees are recognized as public employees of the state.

Additionally, the National Labor Relations Board further identified wide-ranging problems that would arise if joint employers with little actual connection to employees were subjected to bargaining with a union representing the joint employees. Moreover, it would lead to conflicts with franchise law, trademark law, and even current NLRB law.