  We establish a homology relation for the Deligne-Mumford moduli spaces of
real curves which lifts to a WDVV-type relation for real Gromov-Witten
invariants of real symplectic manifolds; we also obtain a vanishing theorem for
these invariants. For many real symplectic manifolds, these results reduce all
genus 0 real invariants with conjugate pairs of constraints to genus 0
invariants with a single conjugate pair of constraints. In particular, we give
a complete recursion for counts of real rational curves in odd-dimensional
projective spaces with conjugate pairs of constraints and specify all cases
when they are nonzero and thus provide non-trivial lower bounds in
high-dimensional real algebraic geometry. We also show that the real invariants
of the three-dimensional projective space with conjugate point constraints are
congruent to their complex analogues modulo 4.
