Random field with paths given as restrictions of holomorphic functions to Euclidean space-time can be Wick-rotated by pathwise analytic continuation. Euclidean symmetries of the correlation functions then go over to relativistic symmetries. As a concrete example, convoluted point processes with interactions motivated from quantum field theory are discussed. A general scheme for the construction of Euclidean invariant infinite volume measures for systems of continuous particles with ferromagnetic interaction is given and applied to the models under consideration. Connections with Euclidean quantum field theory, Widom-Rowlinson and Potts models are pointed out. For the given models, pathwise analytic continuation and analytically continued correlation functions are shown to exist and to expose relativistic symmetries.