  For gauged scalar fields minimally coupled to Einstein-Maxwell theory, the
Mayo-Bekenstein no-hair theorem can be circumvented when including appropriate
scalar self-interactions, allowing static, electrically charged black holes to
be endowed with (Abelian) gauged scalar hair. Here we show these spherically
symmetric solutions can be extended to include a magnetic charge in a model
with scalar multiplets. The resulting dyonic configurations share most of the
properties of the electrically charged solutions, in particular satisfying the
same {\it resonance} condition, with the existence of a mass gap with respect
to the bald Reissner-Nordstr\"om dyonic black holes. A distinctive feature,
however, is that no solitonic limit exists for a non-zero magnetic charge.
