  We consider a sequence of approximate solutions to the compressible Euler
system admitting uniform energy bounds and/or satisfying the relevant field
equations modulo an error vanishing in the asymptotic limit. We show that such
a sequence either (i) converges strongly in the energy norm, or (ii) the limit
is not a weak solution of the associated Euler system. This is in sharp
contrast to the incompressible case, where (oscillatory) approximate solutions
may converge weakly to solutions of the Euler system. Our approach leans on
identifying a system of differential equations satisfied by the associated
turbulent defect measure and showing that it only has a trivial solution.
