  We propose a second-order accurate semi-implicit and well-balanced finite
volume scheme for the equations of ideal magnetohydrodynamics (MHD) including
gravitational source terms. The scheme treats all terms associated with the
acoustic pressure implicitly while keeping the remaining terms part of the
explicit sub-system. This semi-implicit approach makes the method particularly
well suited for problems in the low Mach regime. We combine the semi-implicit
scheme with the deviation well-balancing technique and prove that it maintains
equilibrium solutions for the magnetohydrostatic case up to rounding errors. In
order to preserve the divergence-free property of the magnetic field enforced
by the solenoidal constraint, we incorporate a constrained transport method in
the semi-implicit framework. Second order of accuracy is achieved by means of a
standard spatial reconstruction technique with total variation diminishing
(TVD) property, and by an asymptotic preserving (AP) time stepping algorithm
built upon the implicit-explicit (IMEX) Runge-Kutta time integrators. Numerical
tests in the low Mach regime and near magnetohydrostatic equilibria support the
low Mach and well-balanced properties of the numerical method.
