Embarrassingly, in Korean, there is a slang word for this kind of relationship between guys.
Might be translated as, "the husband of my wife's sister but only by the hole" ?
I don't know how can I call the husband of my wife's sister in English. Korean word is "Dong-suh 동서(同壻)". That is, Korean people don't say "Hank is the husband of Mr. White's wife's sister" but "Hank is the Mr. White's dong-suh".
With this kinship terminology, if some Korean guys found out that they have slept with the same woman (in many cases a prostitute or a professional), they might jokingly call each other "구멍(hole)동서(the husband of my wife's sister)". It has f-word level vulgarism.
Is there similar slang word/expression in English?
UPDATED
I forgot "brother-in-law". Original words is something like "brother-in-hole". In here, 'Hole' symbolizes the vagina. So I mentioned vulgarism.