  We study the surface quasi-geostrophic equation driven by a generic additive
noise process $W$. By means of convex integration techniques, we establish
existence of weak solutions whenever the stochastic convolution $z$ associated
with $W$ is well defined and fulfills certain regularity constraints.
Quintessentially, we show that the so constructed solutions to the non-linear
equation are controlled by $z$ in a linear fashion. This allows us to deduce
further properties of the so constructed solutions, without relying on
structural probabilistic properties such as Gaussianity, Markovianity or a
martingale property of the underlying noise $W$.
