  This work establishes a structure theorem for compact K\"ahler manifolds with
semipositive anticanonical bundle. Up to finite \'etale cover, it is proved
that such manifolds split holomorphically and isometrically as a product of
Ricci flat varieties and of rationally connected manifolds. The proof is based
on a characterization of rationally connected manifolds through the non
existence of certain twisted contravariant tensor products of the tangent
bundle, along with a generalized holonomy principle for pseudoeffective line
bundles. A crucial ingredient for this is the characterization of uniruledness
by the property that the anticanonical bundle is not pseudoeffective.
