  In this article, we consider the Floer cohomology (with $\Z_2$ coefficients)
between torus fibers and the real Lagrangian in Fano toric manifolds. We first
investigate the conditions under which the Floer cohomology is defined, and
then develop a combinatorial description of the Floer complex based on the
polytope of the toric manifold. We show that if the Floer cohomology is
defined, and the Floer cohomology of the torus fiber is non-zero, then the
Floer cohomology of the pair is non-zero. We use this result to develop some
applications to non-displaceability and the minimum number of intersection
points under Hamiltonian isotopy.
