Recent advances in generative speech models have made it increasingly difficult to distinguish authentic from synthetic audio, enabling new forms of fraud and misinformation. Audio watermarking offers a promising defense by embedding an imperceptible signal into generated speech that can later be detected to verify its provenance. However, recent studies have shown that existing post-hoc watermarking methods fail under neural codecs and denoisers, transformations routinely applied during real-world storage, transmission, and processing, severely limiting their practical utility. Here we introduce CRAW, a codec-robust audio watermarking framework that jointly improves robustness against neural re-synthesis while maintaining high perceptual quality. CRAW combines distortion-aware training with an attention-based pooling mechanism, inference-time perceptual mask- ing, and an error-correcting code to recover the fidelity lost during robust training. Experiments demonstrate that CRAW achieves state-of-the-art robustness against neural codecs, denoisers, and vocoders while maintaining perceptual quality comparable to existing post-hoc watermarking methods. The code is available at https://github.com/DavidC1212/craw.

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