We discuss Gaussian generalized random fields indexed by smooth sections of vector bundles with respect to Markov properties. We propose a new set-up which is suitable for the present question and within which new phenomena are detected naturally. In particular, we give a counterexample to the belief that locality in the RKHS implies the germ Markov property. We also prove the close connection between the Markov property and cokernels of local operators. Furthermore, we prove the Markov property for a very degenerate Gaussian random field.