We prove using symplectic field theory that if the suspension of a hyperbolic
diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled
symplectic six-manifold, then its image contains the boundary of a symplectic
disc with vanishing Maslov index. This prevents such a Lagrangian submanifold
to be monotone, for instance the real locus of a smooth real Fano manifold. It
also prevents any Sol manifold to be in the real locus of an orientable real
Del Pezzo fibration over a curve, confirming an expectation of J. Koll\'ar.
Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic
four-manifolds.