The Chekanov torus in S2 x S2 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 with minimal Maslov number at least two 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 T-Chek in S-2 x S-2, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus T-Clif, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.