  We establish a formula for the Gromov-Witten-Welschinger invariants of
$\mathbb CP^3$ with mixed real and conjugate point constraints. The method is
based on a suggestion by J. Koll\'ar that, considering pencils of quadrics,
some real and complex enumerative invariants of $\mathbb CP^3$ could be
computed in terms of enumerative invariants of $\mathbb CP^1\times\mathbb CP^1$
and of elliptic curves.
