We show that the description of the holomorphic projective 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 projective bundle, and is consequently an affine space. Finally, we study the map which to a branching class associates the corresponding projective bundle with section.