  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.
