Geometric singularities of regular surfaces with nef anti-canonical divisors over imperfect fields

Let $S$ be a regular projective surface over a field $k$ of characteristic $p>0$, with $H^0(S,\mathcal{O}_S)=k$ and $-K_S$ nef. We prove that $S$ is geometrically integral over $k$ when $p\geq 7$, and we also find an example of $S$ that is not geometrically integral when $p=5$.