Place the tripe in a large pot and add cold water to cover. Bring to a boil over high heat. Drain and repeat. Drain again. Return the tripe to the pot and again add cold water to cover. Bring to a boil over medium-low heat. Add the salt and simmer until the tripe is fork-tender, about 3 hours. Drain, rinse under cold water, then cut the tripe into 1/2-inch strips.
Heat the olive oil in a large pan over medium-low heat. When the olive oil begins to shimmer, add the onion and carrot. Season with a pinch of salt. Cook, stirring, until the onion and carrot are softened, about 15 minutes. Add the wine and cook until the alcohol aroma dissipates, about 2 minutes. Add the tomatoes and the sliced tripe and cook for about 1 hour. Turn off the heat, add the mint and 1 cup of Pecorino Romano, and mix well. Season to taste.
Plate and sprinkle each portion with the remaining Pecorino Romano. Serve immediately.
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