How non-Gibbsianness helps a metastable Morita minimizer to provide a stable free energy

We analyze a simple approximation scheme based on the Morita-approach for the example of the mean field random field Ising model where it is claimed to be exact in some of the physics literature. We show that the approximation scheme is flawed, but it provides a set of equations whose metastable solutions surprisingly yield the correct solution of the model. We explain how the same equations appear in a different way as rigorous consistency equations. We clarify the relation between the validity of their solutions and the almost surely discontinuous behavior of the single-site conditional probabilities.