  We show that the description of the holomorphic $\mathbb C \mathrm
P^1$-bundle associated to a holomorphic projective structure on a Riemann
surface in terms of the principal bundle of projective $2$-frames extends very
well to the setting of branched projective structures. This generalization
reveals a space of parameters, each of which is associated to a branching
class. The space of branched projective structures with a given branching class
appears as a space of connections on a given $\mathbb C \mathrm P^1$-bundle,
and is consequently an affine space. Finally, we study the map which to a
branching class associates the corresponding $\mathbb C \mathrm P^1$-bundle
with section.
