  Coherences in mutually unbiased bases of states of an isolated quantum system
follow a complementarity relation. The nonlocal advantage of quantum coherence
(NAQC), defined in a bipartite scenario, is a situation in which the average
quantum coherences of the ensembles of one subsystem, effected by a measurement
performed on the other subsystem, violates the complementarity relation. We
analyze two criteria to detect NAQC for bipartite quantum states. We construct
a more generalized version of the criterion to detect NAQC that is better than
the standard criterion as it can capture more states exhibiting NAQC. We prove
the local unitary invariance of these NAQC criteria. Further on, we focus on
investigating the monogamy properties of NAQC in the tripartite scenario. We
check for monogamy of NAQC from two perspectives, differentiated by whether or
not the nodal observer in the monogamy relation performs the measurement for
the nonlocal advantage. We find in particular that in the case where the nodal
observer does not perform the measurement, a strong monogamy relation - an
exclusion principle - is exhibited by NAQC.
