  In this paper, we study the problem of global existence of weak solutions for
the quasi-stationary compressible Stokes equations with an anisotropic viscous
tensor. The key element of our proof is the control of a particular defect
measure associated to the pressure which avoids the use of the eective ux.
Using this new tool, we solve an open problem namely global existence of
solutions {\`a} la Leray for such a system without assuming any restriction on
the anisotropy amplitude. It provides a exible and natural way to treat
compressible quasilinear Stokes systems which are important for instance in
biology, porous media, supra-conductivity or other applications in the low
Reynolds number regime.
