  The orientability problem in real Gromov-Witten theory is one of the
fundamental hurdles to enumerating real curves. In this paper, we describe
topological conditions on the target manifold which ensure that the
uncompactified moduli spaces of real maps are orientable for all genera of and
for all types of involutions on the domain. In contrast to the typical
approaches to this problem, we do not compute the signs of any diffeomorphisms,
but instead compare them. Many projective complete intersections, including the
renowned quintic threefold, satisfy our topological conditions. Our main result
yields real Gromov-Witten invariants of arbitrary genus for real symplectic
manifolds that satisfy these conditions and have empty real locus and
illustrates the significance of previously introduced moduli spaces of maps
with crosscaps. We also apply it to study the orientability of the moduli
spaces of real Hurwitz covers.
