Are there any frogs that keep their tails in adulthood?
Basically no. The famous 'tailed frog', genus Ascaphus, has a tail-like posterior extension of the cloaca (used for internal fertilisation, very rare in frogs), and not a true tail
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There is a pretty good reason adult frogs cannot really keep a tail. Tails are made of post-sacral vertebrae. The sacrum is the vertebra involved in the hip. A tail is separate from the main trunk of the body. This is achieved by the tail starting behind the cloaca, which is basically where the body ends. In frogs, two things are going on.
Firstly, frogs have extended parts of the hip that articulate (interact) with the sacrum—the ilia. This elongation is responsible for frogs being so good at jumping.
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Because the cloaca is kind of tied to the hip, the cloaca of the frog is shifted dramatically backwards relative to the sacrum. So now, the end of the main trunk of the body is not right behind the sacrum like it is in most tetrapods, but shifted waaay back, all the way to where the hip is.
Secondly, all frogs actually have post-sacral vertebrae. But they are fused into a single rod, called the urostyle, which does not extend outside of the body, but is instead coopted into the funky jumpy hip joint. If frogs were to gain a tail, it would either have to be (a) by breaking the urostyle back into its component vertebrae and separating them out of the main body of the animal—extremely unlikely, because it would compromise the hip joint and require major muscle restructuring, or (b) by adding additional vertebrae behind the urostyle that would be normal—extremely unlikely because there are no known mutants that do anything like that to my knowledge, and it would also probably require some kind of support at the base (a second hip?), which is hard to imagine.
So in a way, frogs actually have tails, they're just on the inside. But it is very, very unlikely that any frog would evolve a tail again, without first rebuilding the whole hip mechanism. Even with the weird-ass stuff that evolution has done to frogs over the millennia, they seem to be pretty canalised on this tailless fate. But, at least they get to have a highly functional tail as tadpoles—at least, in those species that have tadpoles.
Anyone wanting to do some hardcore reading on the topic of urostyle ontogeny and evolution can check out this paper on the topic.
