$W^\pm Z$ production at NNLO QCD and NLO EW matched to parton showers with MiNNLO$_{\rm PS}$

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.