  We present a novel quasi-conservative arbitrary high order accurate ADER
discontinuous Galerkin (DG) method allowing to efficiently use a
non-conservative form of the considered partial differential system, so that
the governing equations can be solved directly in the most physically relevant
set of variables. This is particularly interesting for multi-material flows
with moving interfaces and steep, large magnitude contact discontinuities, as
well as in presence of highly non-linear thermodynamics. However, the
non-conservative formulation of course introduces a conservation error which
would normally lead to a wrong approximation of shock waves. Hence, from the
theoretical point of view, we give a formal definition of the conservation
defect of non-conservative schemes and we analyze this defect providing a local
quasi-conservation condition, which allows us to prove a modified Lax-Wendroff
theorem. Then, to deal with shock waves in practice, we exploit the framework
of the so-called a posteriori subcell finite volume (FV) limiter, so that, in
troubled cells appropriately detected, we can incorporate a local conservation
correction. Our corrected FV update entirely removes the local conservation
defect, allowing, at least formally, to fit in the hypotheses of the proposed
modified Lax-Wendroff theorem. Here, the shock-triggered troubled cells are
detected by combining physical admissibility criteria, a discrete maximum
principle and a shock sensor inspired by Lagrangian hydrodynamics.
  To prove the capabilities of our novel 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 by showing
the improved reliability of our scheme on the multi-fluid Euler system on
examples like the interaction of a shock with a helium bubble.
