  Up to finite quasi-\'etale cover, any smooth complex projective variety $X$
with nef anti-canonical bundle is a holomorphic fibre bundle over a $K$-trivial
variety with locally constant transition functions. We consider the converse
problem and prove that (essentially) any fibre bundle with locally constant
transition functions over a $K$-trivial variety has a nef anti-canonical
bundle. We also study the same problem for similar positivity notions and we
study when such fibre bundles are projective. Finally, we complement some
results on the structure theory of varieties whose tangent bundle admits a
singular hermitean metric of positive curvature.
