Unsupervised Domain Adaptation (UDA) person search aims to adapt models trained on labeled source data to unlabeled target domains. Existing approaches typically rely on clustering-based proxy learning, but their performance is often undermined by unreliable pseudo-supervision. This unreliability mainly stems from two challenges: (i) spectral shift bias, where low- and high-frequency components behave differently under domain shifts but are rarely considered, degrading feature stability; and (ii) static proxy updates, which make clustering proxies highly sensitive to noise and less adaptable to domain shifts. To address these challenges, we propose the Reliable Pseudo-supervision in UDA Person Search (RPPS) framework. At the feature level, a Dual-branch Wavelet Enhancement Module (DWEM) embedded in the backbone applies discrete wavelet transform (DWT) to decompose features into low- and high-frequency components, followed by differentiated enhancements that improve cross-domain robustness and discriminability. At the proxy level, a Dynamic Confidence-weighted Clustering Proxy (DCCP) employs confidence-guided initialization and a two-stage online-offline update strategy to stabilize proxy optimization and suppress proxy noise. Extensive experiments on the CUHK-SYSU and PRW benchmarks demonstrate that RPPS achieves state-of-the-art performance and strong robustness, underscoring the importance of enhancing pseudo-supervision reliability in UDA person search. Our code is accessible at https://github.com/zqx951102/RPPS.