Heat a Dutch oven or stockpot over medium-high heat. Cook and stir beef, red bell peppers, green onions, and garlic in the hot Dutch oven until browned and crumbly, 5 to 10 minutes; drain and discard grease. Add water, diced tomatoes, tomato paste, Italian seasoning, salt, and pepper to ground beef mixture; bring to a boil.

Stir farfalle pasta into ground beef-tomato mixture; cook for 10 minutes. Add corn, peas, and mushrooms to mixture and continue cooking, stirring occasionally, until pasta is cooked through yet firm to the bite, about 5 more minutes.