Heat vegetable oil in a large stock pot over medium heat. Add onion and garlic; cook and stir until brown and tender, about 5 minutes. Stir in 2 cans of chicken broth and canned tomatoes. Add fresh tomatoes.
Mix Anaheim chile peppers, jalapeno pepper, cumin, chili powder, and pepper into the pot. Stir in chicken. Bring to a boil; reduce heat to medium and simmer soup until flavors combine, about 20 minutes.
Ladle soup into serving bowls; twist 1 lime wedge into each bowl. Top with corn chips, sour cream, Cheddar cheese, avocado slices, and cilantro.