  We study three-state Potts spins on a square lattice, in which all bonds are
ferromagnetic along one of the lattice directions, and antiferromagnetic along
the other. Numerical transfer-matrix are used, on infinite strips of width $L$
sites, $4 \leq L \leq 14$. Based on the analysis of the ratio of scaled mass
gaps (inverse correlation lengths) and scaled domain-wall free energies, we
provide strong evidence that a critical (Kosterlitz-Thouless) phase is present,
whose upper limit is, in our best estimate, $T_c=0.29 \pm 0.01$. From analysis
of the (extremely anisotropic) nature of excitations below $T_c$, we argue that
the critical phase extends all the way down to T=0. While domain walls parallel
to the ferromagnetic direction are soft for the whole extent of the critical
phase, those along the antiferromagnetic direction seem to undergo a softening
transition at a finite temperature. Assuming a bulk correlation length varying,
for $T>T_c$, as $\xi (T) =a_\xi \exp [ b_\xi (T-T_c)^{-\sigma}]$, $\sigma
\simeq 1/2$, we attempt finite-size scaling plots of our finite-width
correlation lengths. Our best results are for $T_c=0.50 \pm 0.01$. We propose a
scenario in which such inconsistency is attributed to the extreme narrowness of
the critical region.
