Wick rotation to thermal of QFT in Minkowski space to thermal QFT, which is after this transformation analogue to statistical mechanics, does only describe equilibrium statistical mechanics. On page 227 of this paper it is said, that for dynamical questions beyond the equilibrium properties, the time coordinate has to be analytically continued to real Minkowski time.

This I do not understand at all. First of all, does analytic continuation not always generalize a real quantity to a complex quantity? This seems to go the other way round? Can somebody explain a bit why such an analytic continuation allows to describe the evolution of a system and how it works?