  This paper is devoted to the analysis of a singular perturbation problem for
a $2$-D incompressible MHD system with density variations and Coriolis force,
in the limit of small Rossby numbers. Two regimes are considered. The first one
is the quasi-homogeneous regime, where the densities are small perturbations
around a constant state. The limit dynamics is identified as an incompressible
homogeneous MHD system, coupled with an additional transport equation for the
limit of the density variations. The second case is the fully non-homogeneous
regime, where the densities vary around a general non-constant profile. In this
case, in the limit, the equation for the magnetic field combines with an
underdetermined linear equation, which links the limit density variation
function with the limit velocity field. The proof is based on a compensated
compactness argument, which enables us to consider general ill-prepared initial
data. An application of Di Perna-Lions theory for transport equations allows to
treat the case of density-dependent viscosity and resistivity coefficients.
