If you're going to pick up the shrimp by the tail and eat it and then throw the tail away, that's fine. But if you're using shrimp in a dish where you're going to eat it with a knife and fork, it definitely shouldn't have tails on it.
The fibrous part of the crustacean is made up of tough muscle tissue, along with chitin and chitosan, which are responsible for that hardened exterior.
The tail works as a built-in handle when you're using your fingers to munch on shrimp cocktail or navigate a seafood boil.
Leaving the tail intact can help enhance the umami flavors of the shrimp, and it is indeed edible depending on how you prepare it.
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