The classical Corlette--Simpson (CS) correspondence relates local systems on complex varieties to Higgs bundles; it is highly transcendental in nature. Its characteristic p counterpart surprisingly turns out to be purely algebraic: Bezrukavnikov identified de Rham local systems on a smooth variety X over F_p with Higgs bundles twisted by a natural G_m-gerbe on the cotangent bundle T_X. By trivializing the gerbe over suitable loci in T_X using additional choices, Ogus--Vologodsky then recovered an honest CS correspondence (i.e., with untwisted Higgs bundles). In this talk, I'll explain that this story has an exact analog for a smooth rigid space X over a perfectoid p-adic field: (generalized) local systems identify with Higgs bundles twisted by a natural G_m-gerbe on T_X, and honest CS correspondences (as studied by many authors in the last 2 decades) can be recovered by trivializing the gerbe over suitable loci in T_X. This is joint work in progress with Mingjia Zhang, and is inspired by recent work of Heuer.