  We investigate when two generalized real Bott manifolds of height 2 have
isomorphic cohomology rings with Z/2 coefficients and also when they are
diffeomorphic. It turns out that cohomology rings with Z/2 coefficients do not
distinguish those manifolds up to diffeomorphism in general. This gives a
counterexample to the cohomological rigidity problem for real toric manifolds
posed in \cite{ka-ma08}. We also prove that generalized real Bott manifolds of
height 2 are diffeomorphic if they are homotopy equivalent.
