  The Kardar-Parisi-Zhang (KPZ) equation sets the universality class for
growing and roughening of nonequilibrium surfaces without any conservation law
and nonlocal effects. We argue here that the KPZ equation can be generalised by
including a symmetry-permitted nonlocal nonlinear term of active origin that is
of the same order as the one included in the KPZ equation. Including this term,
the 2D active KPZ equation is stable in some parameter regimes, in which the
interface conformation fluctuations exhibit sub-logarithmic or
super-logarithmic roughness, with nonuniversal exponents, giving positional
generalised quasi-long-ranged order. For other parameter choices, the model is
unstable, suggesting a perturbatively inaccessible algebraically rough
interface or positional short-ranged order. Our model should serve as a
paradigmatic nonlocal growth equation.
