In a large pot over medium heat, combine turkey, broth, canned tomatoes, green chiles, fresh tomatoes, onion, garlic, and lime juice. Season with cayenne, cumin, salt, and pepper. Bring to a boil, then reduce heat, and simmer 15 to 20 minutes.
Stir in avocado and cilantro, and simmer 15 to 20 minutes, until slightly thickened. Spoon into serving bowls, and top with shredded cheese.