  In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation,
which is formally given by \begin{equation*}\partial_t h =\frac{\nu}{2}\Delta h
+ \lambda((\partial_1 h)^2 - (\partial_2 h)^2) +
\nu^\frac{1}{2}\xi,\end{equation*} where $\xi$ denotes a noise which is white
in both space and time, and $\lambda$ and $\nu$ are positive constants. Due to
the wild oscillations of the noise and the quadratic nonlinearity, the previous
equation is classically ill-posed. It is not possible to linearise it via the
Cole-Hopf transformation and the pathwise techniques for singular SPDEs (the
theory of Regularity Structures by M. Hairer or the paracontrolled
distributions approach of M. Gubinelli, P. Imkeller, N. Perkowski) are not
applicable. In the present work, we consider a regularised version of aKPZ
which preserves its invariant measure. We show that in order to have
subsequential limits once the regularisation is removed, it is necessary to
suitably renormalise $\lambda$ and $\nu$. Moreover, we prove that, in the
regime suggested by the (non-rigorous) renormalisation group computations of
[D.E. Wolf, "Kinetic roughening of vicinal surfaces'', Phys. Rev. Lett., 1991],
i.e. $\nu$ constant and the coupling constant $\lambda$ converging to $0$ as
the inverse of the square root logarithm, any limit differs from the solution
to the linear equation obtained by simply dropping the nonlinearity in aKPZ.
