  Given a projective structure on a surface $N$, we show how to canonically
construct a neutral signature Einstein metric with non-zero scalar curvature as
well as a symplectic form on the total space $M$ of a certain rank $2$ affine
bundle $M \to N$. The Einstein metric has anti-self-dual conformal curvature
and admits a parallel field of anti-self-dual planes. We show that locally
every such metric arises from our construction unless it is conformally flat.
The homogeneous Einstein metric corresponding to the flat projective structure
on $\mathbb{RP}^2$ is the non-compact real form of the Fubini-Study metric on
$M=\mathrm{SL}(3, \mathrm{R})/\mathrm{GL}(2, \mathrm{R})$. We also show how our
construction relates to a certain gauge-theoretic equation introduced by
Calderbank.
