Synthetic data offers alternatives for data augmentation and sharing. Till date, it remains unknown how to use watermarking techniques to trace and audit synthetic tables generated by tabular diffusion models to mitigate potential misuses. In this paper, we design TabWak, the first watermarking method to embed invisible signatures that control the sampling of Gaussian latent codes used to synthesize table rows via the diffusion backbone. TabWak has two key features. Different from existing image watermarking techniques, TabWak uses self-cloning and shuffling to embed the secret key in positional information of random seeds that control the Gaussian latents, allowing to use different seeds at each row for high inter-row diversity and enabling row-wise detectability. To further boost the robustness of watermark detection against post-editing attacks, TabWak uses a valid-bit mechanism that focuses on the tail of the latent code distribution for superior noise resilience. We provide theoretical guarantees on the row diversity and effectiveness of detectability. We evaluate TabWak on five datasets against baselines to show that the quality of watermarked tables remains nearly indistinguishable from non-watermarked tables while achieving high detectability in the presence of strong post-editing attacks, with a 100% true positive rate at a 0.1% false positive rate on synthetic tables with fewer than 300 rows. Our code is available at the following anonymized repository https://github.com/chaoyitud/TabWak.