In many scientific applications, we observe a system in different conditions in which its components may change, rather than in isolation. In our work, we are interested in explaining the generating process of such a multi-context system using a finite mixture of causal mechanisms. Recent work shows that this causal model is identifiable from data, but is limited to settings where the sparse mechanism shift hypothesis holds and only a subset of the causal conditionals change. As this assumption is not easily verifiable in practice, we study the more general principle that mechanism shifts are independent, which we formalize using the algorithmic notion of independence. We introduce an approach for causal discovery beyond partially directed graphs using Gaussian Process models, and give conditions under which we provably identify the correct causal model. In our experiments, we show that our method performs well in a range of synthetic settings, on realistic gene expression simulations, as well as on real-world cell signaling data.