  We prove global results about actions of cocompact lattices in higher-rank
simple Lie groups on closed manifolds endowed with either a projective class of
connections or a conformal class of pseudo-Riemannian metrics of signature
$(p,q)$, with $\min(p,q) \geq 2$. In the continuity of a recent article,
provided that such a structure is locally equivalent to its model $\mathbf{X}$,
the main question treated here is the completeness of the associated
$(G,\mathbf{X})$-structure. Because of the similarities between the model
spaces of projective geometry and non-Lorentzian conformal geometry, a number
of arguments apply in both contexts. We therefore present the proofs in
parallel. The conclusion is that in both cases, when the real-rank is maximal,
the manifold is globally equivalent to either the model space $\mathbf{X}$ or
its double cover.
