  The Ashkin-Teller model can be represented by a pair $(\tau,\tau')$ of Ising
spin configurations with coupling constants $J$ and $J'$ for each, and $U$ for
their product. We study this representation on the integer lattice
$\mathbb{Z}^d$ for $d\geq 2$. We confirm the presence of a partial
antiferromagnetic phase in the isotropic case ($J=J'$) when $-U>0$ is
sufficiently large and $J=J'>0$ is sufficiently small, by means of a graphical
representation. In this phase, $\tau$ is disordered, admitting exponential
decay of correlations, while the product $\tau\tau'$ is antiferromagnetically
ordered, which is to say that correlations are bounded away from zero but
alternate in sign. No correlation inequalities are available in this part of
the phase diagram. In the planar case $d=2$, we construct a coupling with the
six-vertex model and show, in analogy to the first result, that the
corresponding height function is localised, although with antiferromagnetically
ordered heights on one class of the graph.
  We then return to $d\geq 2$ and consider a part of the phase diagram where
$U<0$ but where correlation inequalities still apply. Using the OSSS
inequality, we proceed to establish a subcritical sharpness statement along
suitable curves covering this part, circumventing the difficulty of the lack of
general monotonicity properties in the parameters. We then address the
isotropic case and provide indications of monotonicity.
