Precipitation extremes are expected to become stronger and more frequent in response to anthropogenic global warming. Accurately projecting the ecological and socioeconomic impacts is an urgent task. Impact models are developed and calibrated with observation-based data but rely on Earth system model (ESM) output for future scenarios. ESMs, however, exhibit significant biases in their output because they cannot fully resolve complex cross-scale interactions of processes that produce precipitation cannot. State-of-the-art bias correction methods only address errors in the simulated frequency distributions, locally at every individual grid cell. Improving unrealistic spatial patterns of the ESM output, which would require spatial context, has not been possible so far.Here we show that a post-processing method based on physically constrained generative adversarial networks (GANs) can correct biases of a state-of-the-art global ESM both in local frequency distributions and in the spatial patterns at once. While our method improves local frequency distributions similarly well to a gold-standard ESM bias-adjustment framework, it strongly outperforms existing methods in correcting spatial patterns. Our study highlights the importance of physical constraints in neural networks for out-of-sample predictions in the context of climate change.