Multisite medical data sharing is critical in modern clinical practice and medical research. The challenge is to conduct data sharing that preserves individual privacy and data utility. The shortcomings of traditional privacy-enhancing technologies mean that institutions rely upon bespoke data sharing contracts. The lengthy process and administration induced by these contracts increases the inefficiency of data sharing and may disincentivize important clinical treatment and medical research. This paper provides a synthesis between 2 novel advanced privacy-enhancing technologies-homomorphic encryption and secure multiparty computation (defined together as multiparty homomorphic encryption). These privacy-enhancing technologies provide a mathematical guarantee of privacy, with multiparty homomorphic encryption providing a performance advantage over separately using homomorphic encryption or secure multiparty computation. We argue multiparty homomorphic encryption fulfills legal requirements for medical data sharing under the European Union's General Data Protection Regulation which has set a global benchmark for data protection. Specifically, the data processed and shared using multiparty homomorphic encryption can be considered anonymized data. We explain how multiparty homomorphic encryption can reduce the reliance upon customized contractual measures between institutions. The proposed approach can accelerate the pace of medical research while offering additional incentives for health care and research institutes to employ common data interoperability standards.