  We study a spontaneously broken Einstein-Yang-Mills-Higgs model coupled via a
Higgs portal to an uncharged scalar $\chi$. We present a phase diagram of
self-gravitating solutions showing that, depending on the choice of parameters
of the $\chi$ scalar potential and the Higgs portal coupling constant $
\gamma$, one can identify different regions: If $\gamma$ is sufficiently small
a $\chi$ halo is created around the monopole core which in turn surrounds a
black-hole. For larger values of $\gamma$ no halo exists and the solution is
just a black hole-monopole one. When the horizon radius grows and becomes
larger than the monopole radius solely a black hole solution exists. Because of
the presence of the $\chi$ scalar a bound for the Higgs potential coupling
constant exists and when it is not satisfied, the vacuum is unstable and no
non-trivial solution exists. We briefly comment on a possible connection of our
results with those found in recent dark matter axion models.
