  It has already been shown that in thermal nonlocal Nambu-Jona-Lasinio models
some unphysical behavior, such as a negative pressure, may arise. In this
article, it is shown how this behavior can be related to the pressence of
highly unstable poles of the propagator of the model, for both the Gaussian and
Lorentzian regulators cases. Computations are carried out within the real time
formalism, which allows to isolate the contributions from different poles and
identify the source of these instabilities. It has also been shown in recent
articles how these instabilities are softened by the inclusion of the Polyakov
loop when a Gaussian regulator is considered. This article shows how the
softening of instabilities can be understood by studying the effect of the
Polyakov loop on the poles of the propagator for the Gaussian regulator.
