  In this paper, we propose a definition of genus one real Gromov-Witten
invariants for certain symplectic manifolds with real a structure, including
Calabi-Yau threefolds, and use equivariant localization to calculate certain
genus one real invariants of the projective space. For this definition, we
combine three moduli spaces corresponding to three possible types of
involutions on a symplectic torus, by gluing them along common boundaries, to
get a moduli space without codimension-one boundary and then study orientation
of the total space. Modulo a technical conjectural lemma, we can prove that the
result is an invariant of the corresponding real symplectic manifold. In the
aforementioned example, our main motivation is to show that the physicists
expectation for the existence of separate Annulus, Mobius, and Klein bottle
invariants may not always be true.
