  We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given
by \begin{equation*}
  \partial_t H=\frac12\Delta H+\lambda((\partial_1 H)^2-(\partial_2
H)^2)+\xi\,,
  \end{equation*} where $\xi$ is a space-time white noise and $\lambda$ is a
strictly positive constant. While the classical two-dimensional KPZ equation,
whose nonlinearity is $|\nabla H|^2=(\partial_1 H)^2+(\partial_2 H)^2$, can be
linearised via the Cole-Hopf transformation, this is not the case for AKPZ. We
prove that the stationary solution to AKPZ (whose invariant measure is the
Gaussian Free Field) is superdiffusive: its diffusion coefficient diverges for
large times as $\sqrt{\log t}$ up to $\log\log t$ corrections, in a Tauberian
sense. Morally, this says that the correlation length grows with time like
$t^{1/2}\times (\log t)^{1/4}$. Moreover, we show that if the process is
rescaled diffusively ($t\to t/\varepsilon^2, x\to x/\varepsilon,
\varepsilon\to0$), then it evolves non-trivially already on time-scales of
order approximately $1/\sqrt{|\log\varepsilon|}\ll1$. Both claims hold as soon
as the coefficient $\lambda$ of the nonlinearity is non-zero. These results are
in contrast with the belief, common in the mathematics community, that the AKPZ
equation is diffusive at large scales and, under simple diffusive scaling,
converges the two-dimensional Stochastic Heat Equation (2dSHE) with additive
noise (i.e. the case $\lambda=0$).
