We show that a radially symmetric, nonresonantly pumped spinor exciton-polariton condensate can acquire chirality without a rotating drive, chiral geometry, or pump orbital angular momentum. Spin relaxation in a two-reservoir system shifts the reservoir-induced blueshift relative to the gain profile, creating an effective non-Hermitian chiral potential. A reduced angular-mode theory reveals tunable exceptional points and non-reciprocal coupling between counter-rotating modes. Full driven-dissipative Gross-Pitaevskii simulations show that this hidden chirality enables all-optical formation of spin-selective half-vortices.