  Chirality represents a kind of symmetry breaking characterized by the
noncoincidence of an object with its mirror image and has been attracting
intense attention in a broad range of scientific areas. The recent realization
of spin-orbit coupling in ultracold atomic gases provides a new perspective to
study quantum states with chirality. In this Letter, we demonstrate that the
combined effects of spin-orbit coupling and interatomic soft-core long-range
interaction can induce an exotic supersolid phase in which the chiral symmetry
is broken with spontaneous emergence of circulating particle current. This
implies that a finite angular momentum can be generated with neither rotation
nor effective magnetic field. The direction of the angular momentum can be
altered by adjusting the strength of spin-orbit coupling or interatomic
interaction. The predicted chiral supersolid phase can be experimentally
observed in Rydberg-dressed Bose-Einstein condensates with spin-orbit coupling.
