  For a punctured surface $S$, we characterize the representations of its
fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the
monodromy of a meromorphic projective structure on $S$ with poles of order at
most two and no apparent singularities. This proves the analogue of a theorem
of Gallo-Kapovich-Marden concerning $\mathbb{C}\mathrm{P}^1$-structures on
closed surfaces, and settles a long-standing question about characterizing
monodromy groups for the Schwarzian equation on punctured spheres. The proof
involves a geometric interpretation of the Fock-Goncharov coordinates of the
moduli space of framed $\mathrm{PSL}_2 (\mathbb{C})$-representations, following
ideas of Thurston and some recent results of Allegretti-Bridgeland.
