In a large, heavy pot, heat oil over moderate high heat. Add half the tortilla strips and cook, stirring, until pale golden, about 1 minute. Remove with a slotted spoon and drain on paper towels. Repeat with remaining tortilla strips. Set cooked strips aside in warm oven.
Reduce heat to moderate low. Add onion, garlic and spices; cook, stirring for 5 minutes. Add broth, tomatoes, corn, bay leaves, salt, 1/4 cup cilantro, and one third of the tortilla strips. Bring to a simmer. Cook uncovered for 30 minutes; remove bay leaves. Add chicken and bring soup back to simmer; cook until just done, about 5 to 10 minutes.
To serve, put the remaining tortilla strips in bowls, top with avocado and cheese, and pour in soup. Sprinkle with chopped cilantro. Serve with lime wedges.