We uncover a new global anomaly in two-dimensional sigma models with Majorana–Weyl fermions, arising from the topology of the mapping space into the target manifold G. This geometric phase in the fermion path integral modifies the usual quantization of the Wess–Zumino level, providing a two-dimensional analogue of the parity anomaly—but with a continuously shifted, rather than half-integer, spectrum. Furthermore, our results support the consistency of the low-energy description of N=1 supersymmetric Yang-Mills theory on an interval, by highlighting a subtle and qualitatively distinct nature of the sigma-model anomalies.