  For supercosets with isometry group of the form $\hat{G} \times \hat{G}$, the
eta-deformation can be generalised to a two-parameter integrable deformation
with independent $q$-deformations of the two copies. We study its
kappa-symmetry and write down a formula for the Ramond-Ramond fluxes. We then
focus on $\hat{G}= PSU(1,1|2)$ and construct two supergravity backgrounds for
the two-parameter integrable deformation of the $AdS_3 \times S^3 \times T^4$
superstring, as well as explore their limits. We also construct backgrounds
that are solutions of the weaker generalised supergravity equations of motion
and compare them to the literature.
