Sauté onion, green chilies, and garlic in oil until soft.
Add tomatoes, beef broth, chicken broth, water, tomato juice, and seasonings. Bring soup to a boil; lower heat and add chicken. Simmer covered 1 hour.
To serve, place corn tortilla chips in bottom of soup bowl along with 4-5 pieces chicken, 4 small slices avocado, and grated sharp Cheddar cheese. Add soup, and then add a bit more cheese on top. Serve hot.
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