Given a six functor formalism, one can parametrise the ways to twist the f_!-functors a bit without changing f^*. Using the same principles, one obtains general constructions of monodromic and gerbe-twisted sheaves. The usual construction of six functor formalisms via factorisation into classes I and P by Liu-Zheng assumes that morphism in the intersection of I and P are truncated. In joint work in progress with Will Fisher, we show that without this assumption, one can still construct a twisted six functor formalism.