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Can a given Lagrangian submanifold be realized as the fixed point set of ananti-symplectic involution? If so, it is called \emph{real}. We give anobstruction for a closed Lagrangian submanifold to be real in terms of thedisplacement energy of nearby Lagrangians. Applying this obstruction to toricfibres, we obtain that the central fibre of many (and probably all) toricmonotone symplectic manifolds is real only if the corresponding moment polytopeis centrally symmetric. Furthermore, we embed the Chekanov torus in all toricmonotone symplectic manifolds and show that it is exotic and not real,extending Kim's result (arXiv:1909.09972) for $S^2 \times S^2$. Inside productsof $S^2$, we show that all products of Chekanov tori are pairwise distinct andnot real either. These results indicate that real tori are rare. Our methodsare elementary in the sense that we do not use~$J$-holomorphic curves. Instead,we rely on symplectic reduction and the displacement energy of product tori in$\mathbb{R}^{2n}$.

Real Lagrangian Tori and Versal Deformations