  The complex projective structures considered is this article are compact
curves locally modeled on $\mathbb{CP}^1$. To such a geometric object, modulo
marked isomorphism, the monodromy map associates an algebraic one: a
representation of its fundamental group into
$\operatorname{PGL}(2,\mathbb{C})$, modulo conjugacy. This correspondence is
neither surjective nor injective. Nonetheless, it is a local diffeomorphism
[Hejhal, 1975]. We generalize this theorem to projective structures admitting
poles (without apparent singularity and with fixed residues): the corresponding
monodromy map (including Stokes data) is a local biholomorphism.
