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People who go out of their way to make sure not to include "The" when referring to them. Yes, I know they are just called "Eels" and not "The Eels", but that doesn't mean the word "the" should not ever appear in the same sentence with their name. (Although this seems to be an issue with a lot of bands that have a plural noun as their name, the word "the" just seems to run more smoothly with some than with others).
Agreed, the only time I don't find being a stickler for that sort of thing annoying is when it's about, say, tagging them on last.fm. I wouldn't say "Eels are my favorite band" any more than I would say "Hey, can I borrow your The Beatles albums?", both just sound sort of silly and awkward to me.