  We classify the supersymmetric solutions of minimal $N=2$ gauged supergravity
in four dimensions with neutral signature. They are distinguished according to
the sign of the cosmological constant and whether the vector field constructed
as a bilinear of the Killing spinor is null or non-null. In neutral signature
the bilinear vector field can be spacelike, which is a new feature not arising
in Lorentzian signature. In the $\Lambda<0$ non-null case, the canonical form
of the metric is described by a fibration over a three-dimensional base space
that has $\text{U}(1)$ holonomy with torsion. We find that a generalized
monopole equation determines the twist of the bilinear Killing field, which is
reminiscent of an Einstein-Weyl structure. If, moreover, the electromagnetic
field strength is self-dual, one gets the Kleinian signature analogue of the
Przanowski-Tod class of metrics, namely a pseudo-hermitian spacetime determined
by solutions of the continuous Toda equation, conformal to a scalar-flat
pseudo-K\"ahler manifold, and admitting in addition a charged conformal Killing
spinor. In the $\Lambda<0$ null case, the supersymmetric solutions define an
integrable null K\"ahler structure. In the $\Lambda>0$ non-null case, the
manifold is a fibration over a Lorentzian Gauduchon-Tod base space. Finally, in
the $\Lambda>0$ null class, the metric is contained in the Kundt family, and it
turns out that the holonomy is reduced to ${\rm Sim}(1)\times{\rm Sim}(1)$.
There appear no self-dual solutions in the null class for either sign of the
cosmological constant.
