Locally constant fibrations and positivity of curvature

Up to finite étale cover, any smooth complex projective variety $X$ with nef anti-canonical bundle is a holomorphic fibre bundle over a smooth projective variety with trivial canonical class (K-trivial variety for short) with locally constant transition functions. We show that this result is optimal by proving that any projective fibre bundle with locally constant transition functions over a $K$-trivial variety has a nef anti-canonical bundle. Moreover, we complement some results on the structure theory of varieties whose tangent bundle admits a singular Hermitian metric of positive curvature.