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In this paper, we study almost nef regular foliations. We give a structuretheorem of a smooth projective variety $X$ with an almost nef regular foliation$\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ withrationally connected fibers such that $\mathcal{F}$ is a pullback of anumerically flat regular foliation on $Y$. Moreover, $f$ is characterized as arelative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary,an almost nef tangent bundle of a rationally connected variety is genericallyample. For the proof, we generalize Fujita's decomposition theorem. As aby-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sumof a hermitian flat vector bundle and a generically ample reflexive sheaf forany algebraic fiber space $f : X \rightarrow Y$. We also study foliations withnef anti-canonical bundles.

Almost nef regular foliations and Fujita's decomposition of reflexive sheaves