  Given a compact connected Riemann surface $X$ equipped with an
antiholomorphic involution $\tau$, we consider the projective structures on $X$
satisfying a compatibility condition with respect to $\tau$. For a projective
structure $P$ on $X$, there are holomorphic connections and holomorphic
differential operators on $X$ that are constructed using $P$. When the
projective structure $P$ is compatible with $\tau$, the relationships between
$\tau$ and the holomorphic connections, or the differential operators,
associated to $P$ are investigated. The moduli space of projective structures
on a compact oriented $C^\infty$ surface of genus $g\, \geq\, 2$ has a natural
holomorphic symplectic structure. It is known that this holomorphic symplectic
manifold is isomorphic to the holomorphic symplectic manifold defined by the
total space of the holomorphic cotangent bundle of the Teichm\"uller space
${\mathcal T}_g$ equipped with the Liouville symplectic form. We show that
there is an isomorphism between these two holomorphic symplectic manifolds that
is compatible with $\tau$.
