Suppose I have some closed $4$-manifold $X$ and a codimension-two foliation $\mathcal{F}$, as well as a closed surface $\Sigma$ of nonnegative self-intersection that is everywhere transverse to $\mathcal{F}$.

Then what kind of restrictions are there on the foliation $\mathcal{F}$? This question gives some answers in the case where $X$ is a complex surface and $\mathcal{F}$ is holomorphic, but I'm more interested in what happens in the real case.