  We construct spherically symmetric dyonic black holes in a generalized
Maxwell-Friedberg-Lee-Sirlin type model with a complex scalar doublet and a
symmetry breaking potential {for the real scalar field}, minimally coupled to
Einstein gravity in asymptotically flat space.
  We analyze the properties of the hairy black holes and determine their domain
of existence. Our discussion focuses mostly on the case of a long-ranged
massless real scalar field.
  Our results indicate that in this case, depending on the coupling constants,
the resonant hairy dyonic black holes may bifurcate from Reissner-Nordstr\"om
black holes at maximal chemical potential, while the limiting solutions at
minimal chemical potential may be related to the Penney solution.
