In this talk, we present a novel quasi-conservative high order Discontinuous Galerkin (DG) method able to capture contact discontinuities avoiding any spurious numerical artifacts, thanks to the PDE evolution in primitive variables, while at the same time being strongly conservative on shocks, thanks to a conservative a posteriori subcell Finite Volume (FV) limiter. In particular, in this work we consider ADER-DG schemes on Voronoi meshes, and we treat in primitive variables both the predictor step (devoted to producing a local high order accurate space-time reconstruction) and the corrector step which takes care of the final solution update. This is possible because the algorithm is then supplemented with an a posteriori sub-cell FV limiter, where the solution obtained at the end of each time step is checked for physical admissibility, and on the cells where it is judged to be troubled, it is locally recomputed via a robust second order FV scheme. At this point, we distinguish, by using a criterion based on the divergence of the velocity field, between troubled zones caused by shock discontinuities and all the rest; on shocks we now apply a FV acting on the conservative version of the PDE, while all the rest is still evolved according to the primitive formulation. To prove the capabilities of the proposed approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude the presentation by showing the improved reliability of our scheme on the multi-fluid Euler equations on examples, like the interaction of a shock with a helium bubble, for which we are able to avoid the development of any spurious oscillations.