The chiral anomaly is a fundamental quantum mechanical phenomenon which is of great importance to both particle physics and condensed matter physics alike. In the context of QED, it manifests as the breaking of chiral symmetry in the presence of electromagnetic fields. It is also known that anomalous chiral symmetry breaking can occur through interactions alone, as is the case for interacting one-dimensional systems. In this Letter, we investigate the interplay between these two modes of anomalous chiral symmetry breaking in the context of interacting Weyl semimetals. Using Fujikawa's path integral method, we show that the chiral charge continuity equation is modified by the presence of interactions which can be viewed as including the effect of the electric and magnetic fields generated by the interacting quantum matter. This can be understood further using dimensional reduction and a Luttinger liquid description of the lowest Landau level. These effects manifest themselves in the nonlinear response of the system. In particular, we find an interaction-dependent density response due to a change in the magnetic field as well as a contribution to the nonequilibrium and inhomogeneous anomalous Hall response while preserving its equilibrium value.