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

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