  We study natural additional structures on real algebraic surfaces with
trivial first homology mod 2 of the complexification. If the set of real points
realizes the zero of the second homology mod 2 of the complexification, then
the set of real points is equipped with a pair of opposite orientations and a
Spin structure. If the set of real points realizes the same homology class as
the complexification of a real curve on the surface, then the complement of the
curve in set of real points is equipped a pair of opposite orientations, which
do not extend across the curve, and the whole set of real points is equipped
with a Pin^- structure.
  These constructions are similar to the complex orientations of real algebraic
curves dividing their complexifications and generalize to high dimensions.
