Let $(M,\omega)$ be a symplectic manifold, and assume $\dim_\mathbb{R}(M)>4$. Suppose there exists a real Lagrangian fibration $\pi:M\rightarrow Q$ so that $Q$ becomes an integral affine manifold with singularities. We do not assume the generic fiber of $\pi$ is a torus. Let $\Delta\subset Q$ be the discriminant locus of $\pi$ over which the fibers of $\pi$ are singular, is there any example of such an $M$ so that $\Delta$ is actually a finite set of points?
I know that there is no such $M$ after we imposing enough assumptions on $\pi$, what I really like to see is a positive answer, even for very restrictive examples.