Relying on paired synthetic data, existing learning-based Computational Aberration Correction (CAC) methods are confronted with the intricate and multifaceted synthetic-to-real domain gap, which leads to suboptimal performance in real-world applications. In this paper, in contrast to improving the simulation pipeline, we deliver a novel insight into real-world CAC from the perspective of Unsupervised Domain Adaptation (UDA). By incorporating readily accessible unpaired real-world data into training, we formalize the Domain Adaptive CAC (DACAC) task, and then introduce a comprehensive Real-world aberrated images (Realab) dataset to benchmark it. The setup task presents a formidable challenge due to the intricacy of understanding the target optical degradation domain. To this intent, we propose a novel Quantized Domain-Mixing Representation (QDMR) framework as a potent solution to the issue. Centering around representing and quantizing the optical degradation which is consistent across different images, QDMR adapts the CAC model to the target domain from three key aspects: (1) reconstructing aberrated images of both domains by a VQGAN to learn a Domain-Mixing Codebook (DMC) characterizing the optical degradation; (2) modulating the deep features in CAC model with DMC to transfer the target domain knowledge; and (3) leveraging the trained VQGAN to generate pseudo target aberrated images from the source ones for convincing target domain supervision. Extensive experiments on both synthetic and real-world benchmarks reveal that the models with QDMR consistently surpass the competitive methods in mitigating the synthetic-to-real gap, which produces visually pleasant real-world CAC results with fewer artifacts. Codes and datasets are made publicly available at https://github.com/zju-jiangqi/QDMR.