Yep, it's the locus classicus for the fabled nine-tailed fox 구미호 九尾狐, those creatures that are able to transform themselves into beautiful maidens in order to seduce men and take away their vital energy, thereby supplementing the fox's own yin energy (foxes are active at night, thus thought heavy on the yin side and they can transform into that other creature with yin powers: a woman) by absorbing and thus taking away the yang essence of the male "she" seduces, thereby lengthening her own existence. Fox fairies are popular in Chinese and Korean folklore as clever tricksters.
Like vampires, the specific powers and weaknesses of nine-tailed foxes, and fox-fairies in general, depend somewhat on who is telling the story. And between the collections of supernatural stories collected in northeast Asia over the centuries, there has been no shortage of tellers. Some say that a fox that lives for a thousand years turns into a nine-tailed fox. Some say it sprouts a new tail for every hundred years that it lives. Sometimes it has to eat the hearts of men, other times absorbing (and thus robbing them of their) "essence" through sex or sex-like acts is enough. The Asian folklore idea that foxes wed on days when the sun is shining and rain falls at the same time is famously depicted in Kurosawa's movie Dreams. The storytelling continues into the present with such famous modern 구미호 as 고소영
and 김태희 and other schlock that nobody should be forced to sit through. And that's just in Korea! I won't even get into all the Pu Songling inspired stuff produced in China, let alone the stuff in video games or anime.
There are, of course, critters other than nine-tailed foxes that go around with their tails hidden.
Look at the first word in that 시조 by the 순수한 기생 홍랑. It's got one of those now-you-see-it-now-you-don't tails:
With modernized Korean spelling and a few touch-ups to eliminate archaisms, it looks like this:
멧버들 가려 꺾어 보내노라, 임에게 I pick out, break off and send a mountain willow to my beloved,
주무시는 窓밖에 심어두고 보소서 And pray you plant it outside the window where you sleep--
밤비에 새잎 나거든 나인줄 여기소서 When new leaves sprout in the spring rain, please think of them as me.
뫼 is the original spelling for the local word for mountain now spelled as "메," and there is no word 멧버들, but you can still find this mountainous morpheme, complete with 사이시옷 tail the word 멧돼지.
Yep, Korean has, at least since middle-Korean was spoken during the time of Sejong the Great, always had a fun way of making compound nouns by inserting a ㅅ between the words in question to mark the first one as a modifier of the second. Funny thing is that this ㅅ was never prounounced for the sound value /s/. It has always been pronounced by "tensifying" the initial sound following. So "윗이" comes out sounding like /윈니/ not */위시/.
If you are writing in Korean, you can always insert a 시옷받침 at the end of the first word, but when you are writing in Chinese you can't invent a new graph with a ㅅ appended every time you want to use one. That's why, at least in middle-Korean, we get floating 사이시옷 and indeed, it's this aspect that probably gave rise to the "사이" part of the name.
But in modern Korean, it's considered an anathema to have loose consonants running around. So, while the 시옷 would be visible if it were written in 한글, in mixed script the words are just concatinated. No ㅅ. ^^
You can see this in the modern version:
동짓달 기나긴 밤을 한허리를 베어 내어 To cut the waist from this long deep winter month night
춘풍 이불 아래 서리서리 넣었다가 And place it in coils under the spring-wind[-warm] blanket
어룬 임 오신 날 밤이어든 굽이굽이 펴리라 That, at night on the day that my partner beloved comes, I will spread it with meandering turns
Of course if "동짓달" is written in modern mixed script, it becomes a de-시옷'ized "冬至달" (still pronounced /동짓달/ but not spelled that way).
There's also a distinctly 사이시옷 sounding phenomenon in many places where two Chinese morphemes come together. Generally these are not officially recognized as this phenomenon--and thus written with a 사이시옷 (even when writing in 한글)--but it sure sounds awfully similar. I mean words like 대구 (對句), 사건, 대가, 성과, 등등등. The word 효과 is offically pronounced /효과/, as anyone can confirm by listening to well-trained broadcasting announcers, but listen to the man in the street and you will, likely as not, hear the officially unaccepted /효꽈/.
That's a pretty well hidden tail. But you can see hints.
There's another even sneakier one out there, though. The Sanchon Hunjang likes to call these guys 사이히읗, but as far as I know, nobody else has ever used that name. Listen carefully to what happens at the morpheme boundary when the bound-morpheme 수- (male) comes in contact with free morphemes.
수 + 개 => 수캐
수 + 것 => 수컷
수 + 닭 => 수탉
수 + 돼지 => 수퇘지
A definite ㅎ hiding in there, no?
It's not only 수- that has this hidden tail effect. The counterpart 암- (female) does the exact same thing. Not only these sex-indicators, 안- (inside) also does the same. Behold: 안팎. 머리 also shows it's tail when combined with -가락. And on and on and on.
There are indeed many strange creatures that go about in the moonlight with their tails hidden to trick the unwary. Be careful...