  The local, uncorrelated multiplicative noises driving a second-order, purely
noise-induced, ordering phase transition (NIPT) were assumed to be Gaussian and
white in the model of [Phys. Rev. Lett. \textbf{73}, 3395 (1994)]. The
potential scientific and technological interest of this phenomenon calls for a
study of the effects of the noises' statistics and spectrum. This task is
facilitated if these noises are dynamically generated by means of stochastic
differential equations (SDE) driven by white noises. One such case is that of
Ornstein--Uhlenbeck noises which are stationary, with Gaussian pdf and a
variance reduced by the self-correlation time (\tau), and whose effect on the
NIPT phase diagram has been studied some time ago. Another such case is when
the stationary pdf is a (colored) Tsallis' (q)--\emph{Gaussian} which, being a
\emph{fat-tail} distribution for (q>1) and a \emph{compact-support} one for
(q<1), allows for a controlled exploration of the effects of the departure from
Gaussian statistics. As done before with stochastic resonance and other
phenomena, we now exploit this tool to study--within a simple mean-field
approximation and with an emphasis on the \emph{order parameter} and the
``\emph{susceptibility}''--the combined effect on NIPT of the noises'
statistics and spectrum. Even for relatively small (\tau), it is shown that
whereas fat-tail noise distributions ((q>1)) counteract the effect of
self-correlation, compact-support ones ((q<1)) enhance it. Also, an interesting
effect on the susceptibility is seen in the last case.
