  We prove that a torsion-free sheaf $\mathcal F$ endowed with a singular
hermitian metric with semi-positive curvature and satisfying the minimal
extension property admits a direct-sum decomposition $\mathcal F \simeq
\mathcal U \oplus \mathcal A$ where $\mathcal U$ is a hermitian flat bundle and
$\mathcal A$ is a generically ample sheaf. The result applies to the case of
direct images of relative pluricanonical bundles $f_* \omega_{X/Y}^{\otimes m}$
under a surjective morphism $f\colon X \to Y$ of smooth projective varieties
with $m\geq 2$. This extends previous results of Fujita, Catanese--Kawamata,
and Iwai.
