  We obtain the topological obstructions to existence of a bundle of
irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of
arbitrary dimension and signature and prove that bundles of Clifford modules
are associated to so-called real Lipschitz structures. The latter give a
generalization of spin structures based on certain groups which we call real
Lipschitz groups. In the fiberwise-irreducible case, we classify the latter in
all dimensions and signatures. As a simple application, we show that the
supersymmetry generator of eleven-dimensional supergravity in "mostly plus"
signature can be interpreted as a global section of a bundle of irreducible
Clifford modules if and $\textit{only if}$ the underlying eleven-manifold is
orientable and spin.
