Many of the organisations that produce, store and share healthcare information including hospitals, labs, pharmacies, insurers and device manufacturers are all distrustful of one another, and their ability to ensure the integrity of the data, its resistance to unauthorised alteration, and its verifiable provenance is challenging to guarantee across such fractured systems. The distributed ledger technology known as blockchain cryptographically linked blocks of records that make it difficult to tamper with or alter data is one possible candidate for the infrastructure to support this assurance of data integrity without a central trusted authority. The consensus mechanism is the protocol used to agree on the content of the ledger between distrustful nodes, and its impact on a system's throughput, latency, energy usage, fault tolerance and decentralisation determines its suitability for clinical applications. This paper presents a survey of the existing consensus mechanisms and proposes a healthcare-oriented taxonomy of consensus mechanisms. It defines the work in four directions: taxonomy of consensus families, proof-based, stake-based, voting/Byzantine-fault-tolerant and hybrid consensus families; comparative analysis of their performance, security and decentralisation trade-offs with the blockchain trilemma; mapping healthcare applications onto consensus and permission families (electronic health record sharing, consent management, pharmaceutical supply-chain provenance, clinical trials, and remote monitoring); and open challenges of privacy versus transparency, scalability, regulatory compliance and interoperability. The survey shows that deployments in the healthcare sector are dominated by permissioned, voting-based systems like proof-of-authority and practical Byzantine fault tolerance, which have a higher transaction throughput, faster finality, and low energy cost that fulfil the clinical requirements, compared to energy-intensive proof-of-work. The most important design principle is to match up the consensus mechanism to the level of integrity needed for each application rather than using a single ledger for them all.