Weakly supervised person search simultaneously addresses detection and re-identification tasks without relying on person identity labels. Prototype-based contrastive learning is commonly used to address unsupervised person re-identification. We argue that prototypes suffer from spatial, temporal, and label inconsistencies, which result in their inaccurate representation. In this paper, we propose a novel Consistent Prototype Contrastive Learning (CPCL) framework to address prototype inconsistency. For spatial inconsistency, a greedy update strategy is developed to introduce ground truth proposals in the training process and update the memory bank only with the ground truth features. To improve temporal consistency, CPCL employs a local window strategy to calculate the prototype within a specific temporal domain window. To tackle label inconsistency, CPCL adopts a prototype nearest neighbor consistency method that leverages the intrinsic information of the prototypes to rectify the pseudo-labels. Experimentally, the proposed method exhibits remarkable performance improvements on both the CUHK-SYSU and PRW datasets, achieving an mAP of 90.2% and 29.3% respectively. Moreover, it achieves state-of-the-art performance on the CUHK-SYSU dataset. The code will be available on the project website: https://github.com/JackFlying/cpcl.