  We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in
$\mathbb R^2$ with parameter $\beta\in (0,1)$, an active scalar model
interpolating between SQG ($\beta=1$) and the 2D Euler equations ($\beta=0$) in
vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the
deterministic setting is known, but their uniqueness is open. We show that the
addition of a rough Stratonovich transport noise of Kraichnan type regularizes
the PDE, providing strong existence and pathwise uniqueness of solutions for
initial data $\theta_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$
related to the regularity degree $\alpha$ of the noise and the singularity
degree $\beta$ of the velocity field; in particular, we can cover any $\beta\in
(0,1)$ for suitable $\alpha$ and $p$ and we can reach a suitable ("critical")
threshold. The result also holds in the presence of external forcing $f\in
L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data
of the problem; furthermore, they are well approximated by vanishing viscosity
and regular approximations.
