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Due to the scarcity of dense pixel-level semantic annotations for images recorded in adverse visual conditions, there has been a keen interest in unsupervised domain adaptation (UDA) for the semantic segmentation of such images. UDA adapts models trained on normal conditions to the target adverse-condition domains. Meanwhile, multiple datasets with driving scenes provide corresponding images of the same scenes across multiple conditions, which can serve as a form of weak supervision for domain adaptation. We propose Refign, a generic extension to self-training-based UDA methods which leverages these cross-domain correspondences. Refign consists of two steps: (1) aligning the normal-condition image to the corresponding adverse-condition image using an uncertainty-aware dense matching network, and (2) refining the adverse prediction with the normal prediction using an adaptive label correction mechanism. We design custom modules to streamline both steps and set the new state of the art for domain-adaptive semantic segmentation on several adverse-condition benchmarks, including ACDC and Dark Zurich. The approach introduces no extra training parameters, minimal computational overhead -- during training only -- and can be used as a drop-in extension to improve any given self-training-based UDA method. Code is available at https://github.com/brdav/refign.