  A pair of quantum observables diagonal in the same "incoherent" basis can be
measured jointly, so some coherence is obviously required for measurement
incompatibility. Here we first observe that coherence in a single observable is
linked to the diagonal elements of any observable jointly measurable with it,
leading to a general criterion for the coherence needed for incompatibility.
Specialising to the case where the second observable is incoherent (diagonal),
we develop a concrete method for solving incompatibility problems, tractable
even in large systems by analytical bounds, without resorting to numerical
optimisation. We verify the consistency of our method by a quick proof of the
known noise bound for mutually unbiased bases, and apply it to study emergent
classicality in the spin-boson model of an N-qubit open quantum system.
Finally, we formulate our theory in an operational resource-theoretic setting
involving "genuinely incoherent operations" used previously in the literature,
and show that if the coherence is insufficient to sustain incompatibility, the
associated joint measurements have sequential implementations via incoherent
instruments.
