  We demonstrate that the nontrivial dispersion of a plasma driven by a
high-intensity laser pulse qualitatively affects fundamental nonperturbative
QED processes triggered by the laser pulse even in the case that no electrons
remain in the interaction volume, e.g., due to ponderomotive expulsion. In the
electron-free case this plasma effect is mediated by the response current of
the residual ions as a dispersive effect on the laser propagation. The residual
ions hence act as an effective background to the propagating electromagnetic
laser field. We demonstrate that this has an impact on the fundamental
nonlinear QED process of photon emission by an electron upon absorption of a
large number of laser photons, called nonlinear Compton scattering. In two
exemplary cases we find the ion plasma to suppress and enhance the photon
emission rate by approximately $10\%$ for an intermediate and high laser
intensity, respectively. The latter enhancement has no classical analog.
