The AFL-CIO, the nation’s largest federation of labor unions, is intervening after a dispute broke out between two unions over who should represent workers at a New York multi-arts center.

The federation asked the National Labor Relations Board on April 16 to suspend a representation petition at the Brooklyn Academy of Music until it could settle who has the right to unionize the employees.

The AFL-CIO declined to comment, but the request to the NLRB was signed off on by President Richard Trumka.

A United Auto Workers affiliate—the Technical, Office and Professional Union, Local 2110—originally filed to represent about...