  We develop the quantum harmonic analysis framework in the reducible setting
and apply our findings to polyanalytic Fock spaces. In particular, we explain
some phenomena observed in arXiv:2201.10230 and answer a few related open
questions. For instance, we show that there exists a symbol such that the
corresponding Toeplitz operator is unitary on the analytic Fock space but
vanishes completely on one of the true polyanalytic Fock spaces. This follows
directly from an explicit characterization of the kernel of the Toeplitz
quantization, which we derive using quantum harmonic analysis. Moreover, we
show that the Berezin transform is injective on the set of of Toeplitz
operators. Finally, we provide several characterizations of the
$\mathcal{C}_1$-algebra in terms of integral kernel estimates and essential
commutants.
