Breaking the north-south symmetry: dyonic spinning black holes with synchronized gauged scalar hair

We study stationary clouds of a gauged, complex scalar field on a magnetically (and possibly electrically as well) charged Kerr-Newman black hole (BH). The existence of a magnetic charge $Q_m$ promotes a north-south $\textit{asymmetry}$ of the scalar clouds. This breakdown of the clouds' $\mathbb{Z}_2$-symmetry carries through to the spacetime $\textit{geometry}$ for the non-linear continuation of the clouds: a family of magnetically charged (or dyonic) BHs with synchronized gauged scalar hair, which we construct. Their distinct phenomenology is illustrated by their imaging, exhibiting skewed shadows and lensing. Such hairy BHs could, in principle, result from the superradiant instability of magnetically charged Kerr-Newman BHs, unveiling a dynamical mechanism for creating north-south asymmetric BHs from standard $\mathbb{Z}_2$-symmetric electrovacuum BHs.