Global spinors and orientable five-branes

Fermion fields on an M-theory five-brane carry a representation of the double cover of the structure group of the normal bundle. It is shown that, on an arbitrary oriented Lorentzian six-manifold, there is always an Sp(2) twist that allows such spinors to be defined globally. The vanishing of the arising potential obstructions does not depend on spin structure in the bulk, nor does the six-manifold need to be spin or spin(C). Lifting the tangent bundle to such a generalised spin bundle requires picking a generalised spin structure in terms of certain elements in the integral and modulo-two cohomology of the five-brane world-volume in degrees four and five, respectively.