  A Hermitian metric $\omega$ on a complex manifold is called SKT or
pluriclosed if $dd^c\omega=0$. Let M be a twistor space of a compact,
anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We
prove that in this case M is K\"ahler, hence isomorphic to $\C P^3$ or a flag
space. This result is obtained from rational connectedness of the twistor
space, due to F. Campana. As an aside, we prove that the moduli space of
rational curves on the twistor space of a K3 surface is Stein.
