Verifying the authenticity of AI-generated text has become increasingly important with the rapid advancement of large language models, and unbiased watermarking has emerged as a promising approach due to its ability to preserve output distribution without degrading quality. However, recent work reveals that unbiased watermarks can accumulate distributional bias over multiple generations and that existing robustness evaluations are inconsistent across studies. To address these issues, we introduce UWbench, the first open-source benchmark dedicated to the principled evaluation of unbiased watermarking methods. Our framework combines theoretical and empirical contributions: we propose a statistical metric to quantify multi-batch distribution drift, prove an impossibility result showing that no unbiased watermark can perfectly preserve the distribution under infinite queries, and develop a formal analysis of robustness against token-level modification attacks. Complementing this theory, we establish a three-axis evaluation protocol: unbiasedness, detectability, and robustness, and show that token modification attacks provide more stable robustness assessments than paraphrasing-based methods. Together, UWbench offers the community a standardized and reproducible platform for advancing the design and evaluation of unbiased watermarking algorithms.