The unique band topology of Weyl semimetals provides them with a diverse array of strong photovoltaic response phenomena. Here we explore how plasma screening affects these phenomena. We find that the screening field is highly nonlinear, can contain topological contributions, and can amplify the photoresponse, with the self-consistent response exhibiting the hallmarks of bistability, including hysteresis and catastrophes. The bistability emerges from a crossover from linear to saturated polarization as a function of the field amplitude, because carriers in a Weyl cone have bounded group speed v_F. When time-reversal symmetry is broken, tilted Weyl nodes add a Berry-curvature contribution to the in-plane current, making the nonlinear screening response helicity selective.