Let X be a compact complex manifold such that its canonical bundle is numerically trivial. Assume, additionally, that X is either Moishezon or X is Fujiki with dimension at most four. Using the MMP and classical results in foliation theory, we prove a Beauville–Bogomolov type decomposition theorem for X. We deduce that holomorphic geometric structures of affine type on X are in fact locally homogeneous away from an analytic subset of complex codimension at least two, and that they cannot be rigid unless X is an étale quotient of a compact complex torus. Moreover, we establish a characterization of torus quotients using the vanishing of the first two Chern classes which is valid for any compact complex n-folds of algebraic dimension at least . Finally, we show that a compact complex manifold with trivial canonical bundle bearing a rigid geometric structure must have infinite fundamental group if either X is Fujiki, or X is a threefold, or X is of algebraic dimension at most one.