  Measurement incompatibility is a distinguishing property of quantum physics
and an essential resource for many quantum information processing tasks. We
introduce an approach to verify the joint measurability of measurements based
on phase-space quasiprobability distributions. Our results therefore establish
a connection between two notions of non-classicality, namely the negativity of
quasiprobability distributions and measurement incompatibility. We show how our
approach can be applied to the study of incompatibility-breaking channels and
derive incompatibility-breaking sufficient conditions for bosonic systems and
Gaussian channels. In particular, these conditions provide useful tools for
investigating the effects of errors and imperfections on the incompatibility of
measurements in practice. To illustrate our method, we consider all classes of
single-mode Gaussian channels. We show that pure lossy channels with 50% or
more losses break the incompatibility of all measurements that can be
represented by non-negative Wigner functions, which includes the set of
Gaussian measurements.
