We consider the dynamics of the composition of a rational function with a Galois action, acting on the Berkovich projective line of a non-archimedean field. Such an action can be considered as the analogue of anti-holomorphic map in the non-archimedean setting. It turns out that this new type of maps has a Fatou-Julia theory very similar to that of rational functions. I will also mention applications to skew product dynamics over complex numbers. This is joint work with Richard Birkett and Hongming Nie.