Easy axis antiferromagnets are robust against external magnetic fields of moderate strength. Spin reorientations in strong fields can provide insight into more subtle properties of antiferromagnetic materials, which are often hidden by their high ground-state symmetry. Here, we investigate theoretically effects of curvature in ring-shaped antiferromagnetic achiral anisotropic spin chains in strong magnetic fields. We identify the geometry-governed helimagnetic phase transition above the spin-flop field between vortex and onion states. The curvature-induced Dzyaloshinskii-Moriya interaction results in the spin-flop transition being of first or second order, depending on the ring curvature. Spatial inhomogeneity of the Néel vector in the spin-flop phase generates weak ferromagnetic response in the plane perpendicular to the applied magnetic field. Our paper contributes to the understanding of the physics of curvilinear antiferromagnets in magnetic fields and guides prospective experimental studies of geometrical effects relying on spin-chain-based nanomagnets.