  We study a class of close-packed dimer models on the square lattice, in the
presence of small but extensive perturbations that make them non-determinantal.
Examples include the 6-vertex model close to the free-fermion point, and the
dimer model with plaquette interaction previously analyzed in
\cite{A,AL,GMT17a,GMT17b}. By tuning the edge weights, we can impose a non-zero
average tilt for the height function, so that the considered models are in
general not symmetric under discrete rotations and reflections. In the
determinantal case, height fluctuations in the massless (or `liquid') phase
scale to a Gaussian log-correlated field and their amplitude is a universal
constant, independent of the tilt. When the perturbation strength $\lambda$ is
sufficiently small we prove, by fermionic constructive Renormalization Group
methods, that log-correlations survive, with amplitude $A$ that, generically,
depends non-trivially and non-universally on $\lambda$ and on the tilt. On the
other hand, $A$ satisfies a universal scaling relation (`Haldane' or `Kadanoff'
relation), saying that it equals the anomalous exponent of the dimer-dimer
correlation.
