  In this work we extend the results in [6,32] on the 2D IPM system with
constant viscosity (Atwood number $A_{\mu}=0$) to the case of viscosity jump
($|A_{\mu}|<1$). We prove a h-principle whereby (infinitely many) weak
solutions in $C_tL_{w^*}^{\infty}$ are recovered via convex integration
whenever a subsolution is provided. As a first example, non-trivial weak
solutions with compact support in time are obtained. Secondly, we construct
mixing solutions to the unstable Muskat problem with initial flat interface. As
a byproduct, we check that the connection, established by Sz\'ekelyhidi for
$A_{\mu}=0$, between the subsolution and the Lagrangian relaxed solution of
Otto, holds for $|A_{\mu}|<1$ too. For different viscosities, we show how a
pinch singularity in the relaxation prevents the two fluids from mixing
wherever there is neither Rayleigh-Taylor nor vorticity at the interface.
