  We study the vacua of an $SU(3)\times SU(3)$-symmetric model with a
bifundamental scalar. Structures of this type appear in various gauge theories
such as the Renormalizable Coloron Model, which is an extension of QCD, or the
Trinification extension of the electroweak group. In other contexts, such as
chiral symmetry, $SU(3)\times SU(3)$ is a global symmetry. As opposed to more
general $SU(N)\times SU(N)$ symmetric models, the $N=3$ case is special due to
the presence of a trilinear scalar term in the potential. We find that the most
general tree-level potential has only three types of minima: one that preserves
the diagonal $SU(3)$ subgroup, one that is $SU(2)\times SU(2)\times U(1)$
symmetric, and a trivial one where the full symmetry remains unbroken. The
phase diagram is complicated, with some regions where there is a unique
minimum, and other regions where two minima coexist.
