  In arXiv:math/0605587, 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. 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 this nonorientable
surface. 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 the nonorientable surface is not
homeomorphic to the Klein bottle, and for n<4 when the nonorientable surface is
the Klein bottle. We also derive similar orientability results when the
structure group is SU(n).
