Person Re-Identification (re-ID) aims at retrieving images of the same person taken by different cameras. A challenge for re-ID is the performance preservation when a model is used on data of interest (target data) which belong to a different domain from the training data domain (source data). Unsupervised Domain Adaptation (UDA) is an interesting research direction for this challenge as it avoids a costly annotation of the target data. Pseudo-labeling methods achieve the best results in UDA-based re-ID. They incrementally learn with identity pseudo-labels which are initialized by clustering features in the source re-ID encoder space. Surprisingly, labeled source data are discarded after this initialization step. However, we believe that pseudo-labeling could further leverage the labeled source data in order to improve the post-initialization training steps. In order to improve robustness against erroneous pseudo-labels, we advocate the exploitation of both labeled source data and pseudo-labeled target data during all training iterations. To support our guideline, we introduce a framework which relies on a two-branch architecture optimizing classification in source and target domains, respectively, in order to allow adaptability to the target domain while ensuring robustness to noisy pseudo-labels. Indeed, shared low and mid-level parameters benefit from the source classification signal while high-level parameters of the target branch learn domain-specific features. Our method is simple enough to be easily combined with existing pseudo-labeling UDA approaches. We show experimentally that it is efficient and improves performance when the base method has no mechanism to deal with pseudo-label noise. And it maintains performance when combined with base method that already manages pseudo-label noise. Our approach reaches state-of-the-art performance when evaluated on commonly used datasets, Market-1501 and DukeMTMC-reID, and outperforms the state of the art when targeting the bigger and more challenging dataset MSMT.