  In Bose-Einstein condensates, spin-orbit coupling produces supersolidity. It
is a peculiar state of matter, which, in addition to the superfluid behaviour
of weakly interacting Bose condensates, shows a periodic modulation of its
density typical for crystals and called stripe phase. Here, we report the
fabrication of a new type of samples allowing to achieve room-temperature
supersolidity for a quantum fluid of light. The structure is an optical
microcavity filled with a nematic liquid crystal (LC) sandwiched between two
layers of the organic polymer MeLPPP. We demonstrate the formation of cavity
exciton-polaritons in the presence of Rashba-Dresselhaus spin-orbit coupling
(RDSOC), which is tuned by external voltage controlling the LC birefringence.
In the RDSOC regime, we demonstrate exciton-polariton condensation in the two
distinct degenerate minima of the dispersion. The condensate real space
distribution shows both polarization and density stripes, which stem from the
interference between phase-coherent condensate components characterized by
different wavevectors and polarizations. The possibilities offered by this
platform to tune the particle dispersion and to perform full state tomography,
including time-resolved, open wide perspectives for detailed future studies of
the static and dynamical behaviour of supersolids and of quantum fluids in
presence of SOC and topologically non-trivial bands.
