  Supersolidity - a coexistence of superfluidity and crystalline or amorphous
density variations - has been vividly debated ever since its conjecture. While
the initial focus was on helium-4, recent experiments uncovered supersolidity
in ultra-cold dipolar quantum gases. Here, we propose a new self-bound
supersolid phase in a binary mixture of Bose gases with short-range
interactions, making use of the non-trivial properties of spin-orbit coupling.
We find that a first-order phase transition from a self-bound supersolid stripe
phase to a zero-minimum droplet state of the Bose gas occurs as a function of
the Rabi coupling strength. These phases are characterized using the momentum
distribution, the transverse spin polarization and the superfluid fraction. The
critical point of the transition is estimated in an analytical framework. The
predicted density-modulated supersolid stripe and zero-minimum droplet phase
should be experimentally observable in a binary mixture of $^{39}$K with
spin-orbit coupling.
