I show that real pinor bundles (understood as real vector bundles admitting a global Clifford multiplication) can be defined on a pseudo-Riemannian manifold if and only if the later admits a so called
real Lipschitz structure. I describe the classification of such structures as well as the topological obstructions for their existence, some of which have never been considered before. This allows for a fully general treatment of Dirac operators on such bundles.