  For a symplectic manifold with an anti-symplectic involution having non-empty
fixed locus, we construct a model of the moduli space of real sphere maps out
of moduli spaces of decorated disk maps and give an explicit expression for its
first Stiefel-Whitney class. As a corollary, we obtain a large number of
examples, which include all odd-dimensional projective spaces and many complete
intersections, for which many types of real moduli spaces are orientable. For
these manifolds, we define open Gromov-Witten invariants with no restriction on
the dimension of the manifolds or the type of the constraints if there are no
boundary marked points. If there are boundary marked points, we define the
invariants under some restrictions on the allowed boundary constraints, even
though the moduli spaces are not orientable in these cases.
