Combine diced tomatoes with green chiles, chili beans, salsa, dark and light red kidney beans, black beans, onions, garlic, chili powder, red pepper flakes, black pepper, cumin, taco seasoning mix, and hot sauce in a large soup pot. Bring to a boil, reduce heat to low, and cover pot. Simmer until soup is heated and onions are translucent, about 20 minutes.
Heat a large skillet over medium-high heat. Cook and stir ground turkey in the hot skillet until browned and crumbly, about 10 minutes; drain and discard grease. Stir crumbled turkey into chili.
Cover pot and simmer chili until flavors have blended, about 1 hour.