  Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly
constant real spinor are studied. It is shown that finding such metrics reduces
to solving a fourth order integrable PDE, and some examples are given. The
corresponding twistor space is characterised by existence of a preferred
non-zero real section of $\kappa^{-1/4}$, where $\kappa$ is the canonical line
bundle of the twistor space It is demonstrated that if the parallel spinor is
preserved by a Killing vector, then the fourth order PDE reduces to the
dispersionless Kadomtsev--Petviashvili equation and its linearisation.
Einstein--Weyl structures on the space of trajectories of the symmetry are
characterised by the existence of a parallel weighted null vector.
