Let $X$ be a smooth projective variety and $K_X$ the canonical line bundle. If $K_X$ is nef, then the abundance conjecture predicts that it is semiample, so in particular a multiple $mK_X$ has many sections. If $-K_X$ is nef, then it does not have to be semiample (see examples in reference below). I have the following weaker questions:
Question 1: If $-K_X$ nef, does there exist an integer $m>0$ such that $-mK_X$ effective?
Question 2: If $-K_X$ nef, does there exist $m$ as above, and $H$ ample divisor, such that $-mK_X|_H$ effective?
Of course, if $-K_X$ is big - in the interior of the effective cone, both questions are true, so the issue is really on the boundary. From the paper ''Nef Reduction and Anticanonical Bundles'' I can extract that Bauer-Peternell claim that the result is true in dimensions $\leq3$: see 1.5-1.7 for surfaces, and the various theorems in the case of threefolds. An idea for a counterexample for Q1 would be to cook up a semistable projective bundle, perhaps over something $K$-trivial, and follow something like the Hartshorne-Mumford construction for non-closed effective cone, but it does not seem clear how to make this work in higher dimension or for $-K_X$. Of course, in higher dimension there is more space for a complicated base locus for $-K_X$, which an ample divisor $H$ could perhaps not avoid. Note that one could also ask Q2 for $K_X$ as a weak version of abundance, but even here I am not sure what is known.