  We consider the unique measure of maximal entropy for proper 3-colorings of
$\mathbb{Z}^2$, or equivalently, the so-called zero-slope Gibbs measure. Our
main result is that this measure is Bernoulli, or equivalently, that it can be
expressed as the image of a translation-equivariant function of independent and
identically distributed random variables placed on $\mathbb{Z}^2$. Along the
way, we obtain various estimates on the mixing properties of this measure.
