  We consider admissible weak solutions to the compressible Euler system with
source terms, which include rotating shallow water system and the Euler system
with damping as special examples. In the case of anti-symmetric sources such as
rotations, for general piecewise Lipschitz initial densities and some suitably
constructed initial momentum, we obtain infinitely many global admissible weak
solutions. Furthermore, we construct a class of finite-states admissible weak
solutions to the Euler system with anti-symmetric sources. Under the additional
smallness assumption on the initial densities, we also obtain multiple
global-in-time admissible weak solutions for more general sources including
damping. The basic framework are based on the convex integration method
developed by De~Lellis and Sz\'{e}kelyhidi \cite{dLSz1,dLSz2} for the Euler
system. One of the main ingredients of this paper is the construction of
specified localized plane wave perturbations which are compatible with a given
source term.
