  Earlier we introduced and studied the concept of holomorphic {\it branched
Cartan geometry}. We define here a foliated version of this notion; this is
done in terms of Atiyah bundle. We show that any complex compact manifold of
algebraic dimension $d$ admits, away from a closed analytic subset of positive
codimension, a nonsingular holomorphic foliation of complex codimension $d$
endowed with a transversely flat branched complex projective geometry
(equivalently, a ${\mathbb C}P^d$-geometry). We also prove that transversely
branched holomorphic Cartan geometries on compact complex projective rationally
connected varieties and on compact simply connected Calabi-Yau manifolds are
always flat (consequently, they are defined by holomorphic maps into
homogeneous spaces).
