  We introduce a novel structure-preserving method in order to approximate the
compressible ideal Magnetohydrodynamics (MHD) equations. This technique
addresses the MHD equations using a non-divergence formulation, where the
contributions of the magnetic field to the momentum and total mechanical energy
are treated as source terms. Our approach uses the Marchuk-Strang splitting
technique and involves three distinct components: a compressible Euler solver,
a source-system solver, and an update procedure for the total mechanical
energy. The scheme allows for significant freedom on the choice of Euler's
equation solver, while the magnetic field is discretized using a
curl-conforming finite element space, yielding exact preservation of the
involution constraints. We prove that the method preserves invariant domain
properties, including positivity of density, positivity of internal energy, and
the minimum principle of the specific entropy. If the scheme used to solve
Euler's equation conserves total energy, then the resulting MHD scheme can be
proven to preserve total energy. Similarly, if the scheme used to solve Euler's
equation is entropy-stable, then the resulting MHD scheme is entropy stable as
well. In our approach, the CFL condition does not depend on magnetosonic
wave-speeds, but only on the usual maximum wave speed from Euler's system. To
validate the effectiveness of our method, we solve a variety of ideal MHD
problems, showing that the method is capable of delivering high-order accuracy
in space for smooth problems, while also offering unconditional robustness in
the shock hydrodynamics regime as well.
