  We study the relationship between quantum synchronization and the
thermodynamic performance of a four-level near-degenerate extension of the
Scovil-Schulz Dubois thermal maser. We show how the existence of interacting
coherences can potentially modify the relationship between synchronization and
the coherent power output of such a maser. In particular, the cooperation and
competition between interacting coherences, causes the coherent heat and
efficiency to be bounded by the synchronization measure in addition to the
well-studied power synchronization bound. Overall, our results highlight the
role of quantum synchronization in the working of a thermal machine.
