Steal My Ideas, Please

Finally, got my Internet back, after a few more weeks of problems with Verizon DSL. While working on a new puzzle for word lovers, I came up with this idea: Words that have the same sounds rearranged. Like mix and skim. See? Unlike anagrams, which rearrange letters, these words can use different letters. Must be many in English, and maybe even some that don’t have any letters in common! (psy and ice?) Some anaphones can even be paired in ways that makes sense: chic quiche, gaudy doggy, bright tribe. Ever see this idea before, and is there a word for it? (By the way, it’s possible for two anagrams to have NO sounds in common. At least that’s true of ewe and wee.)