  The Luttinger-Ward functional (LWF) has been a starting point for conserving
approximations in many-body physics for 50 years. The recent discoveries of its
multivaluedness and the associated divergence of the two-particle irreducible
vertex function $\Gamma$ have revealed an inherent limitation of this approach.
Here we demonstrate how these undesirable properties of the LWF can lead to a
failure of computational methods based on an approximation of the LWF. We apply
the Nested Cluster Scheme (NCS) to the Hubbard model and observe the existence
of an additional stationary point of the self-consistent equations, associated
with an unphysical branch of the LWF. In the strongly correlated regime,
starting with the first divergence of $\Gamma$, this unphysical stationary
point becomes attractive in the standard iterative technique used to solve
DMFT. This leads to an incorrect solution, even in the large cluster size
limit, for which we discuss diagnostics.
