Under the Federal Service Labor-Management Relations Statute (external link), 5 USC §§ 7101 ff, exclusive recognition is normally obtained by a union as a result of receiving a majority of votes cast in a representational election. (Exclusive recognitions obtained under Executive Order 10988, which didn't require secret-ballot elections, are preserved via a "grandfather" clause.)
The rights a union is accorded as a result of being certified as the exclusive representative (external link) of the employees in a bargaining unit include, among other things, the right to negotiate bargainable aspects of the conditions of employment of bargaining unit employees, to be afforded an opportunity to be present at formal discussions, to free checkoff arrangements and, at the request of the employee, to be present at Weingarten examinations of unit employees.