The Internet of things (IoT) has made it possible for health institutions to have remote diagnosis, reliable, preventive and real-time decision making. However, the anonymity and privacy of patients are not considered in IoT. Therefore, this paper proposes a blockchain-based anonymous system, known as GarliMediChain, for providing anonymity and privacy during COVID-19 information sharing. In GarliMediChain, garlic routing and blockchain are integrated to provide low-latency communication, privacy, anonymity, trust and security. Also, COVID-19 information is encrypted multiple times before transmitting to a series of nodes in the network. To ensure that COVID-19 information is successfully shared, a blockchainbased coalition system is proposed. The coalition system enables health institutions to share information while maximizing their payoffs. In addition, each institution uses the proposed fictitious play to study the strategies of others in order to update its belief by selecting the best responses from them. Furthermore, simulation results show that the proposed system is resistant to security-related attacks and is robust, efficient, and adaptive. From the results, the proposed proof-of-epidemiology-of-interest (PoEoI) consensus protocol has 15.93% less computational cost than 26.30% of proof-of-work (PoW) and 57.77% proof-ofauthority (PoA) consensus protocol, respectively. Nonetheless, the proposed GarliMediChain system promotes global collaborations by combining existing anonymity and trust solutions with the support of blockchain technology.