Artificial intelligence systems, particularly large language models (LLMs), are increasingly being employed in high-stakes decisions that impact both individuals and society at large, often without adequate safeguards to ensure safety, quality, and equity. Yet LLMs hallucinate, lack common sense, and are biased - shortcomings that may reflect LLMs' inherent limitations and thus may not be remedied by more sophisticated architectures, more data, or more human feedback. Relying solely on LLMs for complex, high-stakes decisions is therefore problematic. Here we present a hybrid collective intelligence system that mitigates these risks by leveraging the complementary strengths of human experience and the vast information processed by LLMs. We apply our method to open-ended medical diagnostics, combining 40,762 differential diagnoses made by physicians with the diagnoses of five state-of-the art LLMs across 2,133 medical cases. We show that hybrid collectives of physicians and LLMs outperform both single physicians and physician collectives, as well as single LLMs and LLM ensembles. This result holds across a range of medical specialties and professional experience, and can be attributed to humans' and LLMs' complementary contributions that lead to different kinds of errors. Our approach highlights the potential for collective human and machine intelligence to improve accuracy in complex, open-ended domains like medical diagnostics.