  We study projective structures on a surface having poles of prescribed
orders. We obtain a monodromy map from a complex manifold parameterising such
structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on
the associated marked bordered surface. We prove that the image of this map is
contained in the union of the domains of the cluster charts. We discuss a
number of open questions concerning this monodromy map.
