In many real-world applications, data is inherently decentralized, necessitating data analysis methods that prioritize privacy while delivering interpretable results. Federated Non-Negative Matrix Factorization (FedNMF) meets this requirement by factorizing latent components from distributed data that cannot be freely shared among clients. A significant challenge in FedNMF arises when clients converge on different solutions due to prolonged independent optimization, leading to drift and incoherent models. While Federated Learning (FL) typically mitigates drift through frequent synchronizations and strong regularization, it often overlooks critical properties of Non-Negative Matrix Factorization, such as permutation invariance. As a result, solutions from FedNMF clients may be misidentified by FL drift as distinct, despite being equivalent. Using an alignment-aware drift, we create coherence through proximal optimization and barycenter aggregation for FedNMF. We analyze the computational complexity of our approach, provide efficient heuristics, and ensure the convergence of our algorithms. On a diverse set of real-world and synthetic datasets, we demonstrate the effectiveness of our methods. Replication Material - doi.org/10.5281/zenodo.14501532