We will have a much better sense of the Hydra Tail protocol once the research paper(s) come out. For now, we can infer its functionality based on some of the design patterns of projects currently building on the Hydra family. For example, SingularityNET's HyperCycle sidechain whitepaper discusses CSL, Plutus, and Hydra integrations, including some talk of Hydra Tail.
In short, Hydra Head is a protocol for validator rings - synchronous, permissioned, high-speed execution environments with CSL snapshots. This is a good way for certain sidechains to interface with Cardano, though others like Hypercycle require that some consensus agents may be partially or completely offline (semi-synchronous or asynchronous). Additional protocols - Hydra Tail - are necessary to account for such agents.
Do note however: the authors of the Hypercycle whitepaper mention that the Hydra Tail protocol is "not yet publicly well specified", and further state the intention "to create a custom version of the Hydra tail protocol
that suits Hypercycle’s needs, incorporating mechanisms and ideas beyond
those that have previous occurred in the Cardano sphere."
TBE when Hydra Tail paper comes out