  Can a given Lagrangian submanifold be realized as the fixed point set of an
anti-symplectic involution? If so, it is called \emph{real}. We give an
obstruction for a closed Lagrangian submanifold to be real in terms of the
displacement energy of nearby Lagrangians. Applying this obstruction to toric
fibres, we obtain that the central fibre of many (and probably all) toric
monotone symplectic manifolds is real only if the corresponding moment polytope
is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric
monotone 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 products
of $S^2$, we show that all products of Chekanov tori are pairwise distinct and
not real either. These results indicate that real tori are rare. Our methods
are 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
$\mathbb{R}^{2n}$.
