The process of consolidating medical records from multiple institutions into one data set makes privacy-preserving record linkage (PPRL) a necessity. Most PPRL approaches, however, are only designed to link records from two institutions, and existing multi-party approaches tend to discard non-matching records, leading to incomplete result sets. In this paper, we propose a new algorithm for federated record linkage between multiple parties by a trusted third party using record-level bloom filters to preserve patient data privacy. We conduct a study to find optimal weights for linkage-relevant data fields and are able to achieve 99.5% linkage accuracy testing on the Febrl record linkage dataset. This approach is integrated into an end-to-end pseudonymization framework for medical data sharing.