Imbroglio is a rather beautiful word describing a complicated, often embarrassing situation. That particular meaning was metaphorically applied in 1818, but since the word's first application in 1750 till then, it meant something more like "jumble". The term, as you can see just by looking at it, is Italian, where it meant "tangle". Now that we know that, we can eliminate the prefix -im, which was a generally useful affix that meant "into", "upon", "in", "on", or just denoted derivation for verbs. The root is broglio, meaning "confusion". That's most likely from Middle French brouiller, which is also the etymon of embroil. Due to a connection between confusion and mixing things up, that's reconstructed as coming from a Proto-Germanic root for "broth", brutha, which would be from Proto-Indo-European bhreu, "to boil".
Adam Aleksic, a rising sophomore studying linguistics and government at Harvard University, has been described as the internet's sixth most famous etymologist. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.