Given a complex projective manifold X and a divisor D with normal crossings, we say that the logarithmic tangent bundle T X (−log D) is R-flat if its pull-back to the normalization of any rational curve contained in X is the trivial vector bundle. If moreover −(K X + D) is nef, then the log canonical divisor K X + D is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber F being a toric variety with boundary divisor D |F .