A smectic liquid-crystal phase made from achiral molecules with bent cores was found to have fluid layers that exhibit two spontaneous symmetry-breaking instabilities: polar molecular orientational ordering about the layer normal and molecular tilt. These instabilities combine to form a chiral layer structure with a handedness that depends on the sign of the tilt. The bulk states are either antiferroelectric-racemic, with the layer polar direction and handedness alternating in sign from layer to layer, or antiferroelectric-chiral, which is of uniform layer handedness. Both states exhibit an electric field-induced transition from antiferroelectric to ferroelectric.