Foliations whose first Chern class is nef

Let $\mathcal{F}$ be a foliation on a projective manifold $X$ with $-K_{\mathcal{F}}$ nef. Assume that either $\mathcal{F}$ is regular, or it has a compact leaf. We prove that there is a locally trivial fibration $f\colon X\to Y$,andafoliation$$\mathcal{G}$$on$Y$with$K_{$\mathcal{G}$} \equiv 0$, such that $\mathcal{F}=f^{-1}(\mathcal{G})$.