We consider a random interface model on the discrete torus with $2n$ sites, obtained from the classical corner flip dynamics but with a weak global perturbation, namely an asymmetry of order $n^{-γ}$ of the direction of growth that switches direction based on the sign of the total area under the interface. The slopes of this model can be viewed as a non-simple exclusion process at half filling with globally dependent rates. We show that, for $γ=1$, the hydrodynamic equation of the empirical density is given by a time concatenation of the viscous Burgers equation and the heat equation. Moreover, for $n$ prime and $γ>\frac{6}{7}$, we establish convergence in law of the equilibrium fluctuations to an infinite-dimensional Ornstein-Uhlenbeck process.