  We revisit the question of the "sign phase transition" for interfering
directed paths with real amplitudes in a random medium. The sign of the total
amplitude of the paths to a given point may be viewed as an Ising order
parameter, so we suggest that a coarse-grained theory for system is a dynamic
Ising model coupled to a Kardar-Parisi-Zhang (KPZ) model. It appears that when
the KPZ model is in its strong-coupling ("pinned") phase, the Ising model does
not have a stable ferromagnetic phase, so there is no sign phase transition. We
investigate this numerically for the case of {\ss}1+1 dimensions, demonstrating
the instability of the Ising ordered phase there.
