Combine broth, canned tomatoes, turkey, fresh tomatoes, onion, green chiles, garlic, and lime juice in a large pot over medium heat; season with cayenne pepper, cumin, salt, and black pepper. Bring to a boil, then reduce heat; simmer 15 to 20 minutes.
Stir in avocado and cilantro; simmer until slightly thickened, 15 to 20 minutes. Serve hot, topped with Monterey Jack cheese.
