(OK, I'm going to close this out for what I think is a better proposal. I'm going to leave this here if anyone wants to discuss register)
In What is the difference between the "formality" and "formal" tags? the difference is said to be that formality is asking about the register of something, and formal is for questions that are requesting answers in a particular register.
I propose that we
- Make formal a synonym of a new tag formal-register
- Make formality a synonym of register
- Make informal-english a synonym of a new tag informal-register
The community wiki answer below is so that we can collaborate on the wording to use for the tag wikis if we decide to go forward with the new tags.
If you agree, up-vote this question. If you see potential issues please leave a comment explaining them or an answer with an alternate proposal. If you think that any part of this proposal should not be done (new tags or synonyms) please down-vote the question.