Abstract
The first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface Ε. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of Ε. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when Ε is not homeomorphic to the Klein bottle, and for n ≤ 3 when Ε is the Klein bottle. We also derive similar orientability results when the structure group is SU(n).