  We study conformal metrics with prescribed Gaussian curvature on surfaces
with conical singularities and geodesic boundary in supercritical regimes.
Exploiting a variational argument, we derive a general existence result for
surfaces with at least two boundary components. This seems to be the first
result in this setting. Moreover, we allow to have conical singularities with
both positive and negative orders, that is cone angles both less and grater
than $2\pi$.
