  Given a separable fibration over an algebraically closed field of positive
characteristic, we study an Iitaka-type inequality for the anticanonical
divisors. We conclude that it holds when the source of the fibration is a
threefold or when the target is a curve. We then give counterexamples in
characteristics $2$ and $3$ for fibrations with "very singular" fibres,
constructed from Tango-Raynaud surfaces.
