The Chekanov torus in $S^2\times S^2$ is not real

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus ${\mathbb{T}}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus ${\mathbb{T}}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.