  We consider $W^\pm Z$ production in hadronic collisions and present
high-precision predictions in QCD and electroweak (EW) perturbation theory
matched to parton showers. To this end, we match next-to-next-to-leading order
QCD corrections to parton showers using the MiNNLO$_{\rm PS}$ method and
consistently combine them with next-to-leading order EW corrections matched to
parton showers. This is the first time such accuracy in the event generation is
achieved for any collider process, and we study in detail the impact of
different choices in the combination of QCD and EW corrections as well as QCD
and QED showers. Spin correlations, interferences and off-shell effects are
retained by considering the full leptonic processes $pp \to
\ell^+\ell^-\ell'^\pm \nu_\ell'$ with $\ell'\neq\ell$ and $\ell'=\ell$ without
approximations, and the matching to QED radiation is performed preserving the
resonance structure of the process. We find that NNLO QCD predictions including
QCD and QED shower effects provide a very good approximation in the bulk-region
of the phase space, while EW effects become increasingly important in the
high-energy tails of kinematic distributions. Our default predictions are in
excellent agreement with recent ATLAS data.
