In a large saucepan, combine chicken, 2 cups cilantro, broth, and water. Bring to a boil, reduce heat, and simmer gently until chicken is just cooked through, about 10 minutes. Transfer chicken to a plate; let cool. Cut into 1/2-inch dice. Strain broth through a fine strainer lined with several layers of cheesecloth; reserve. Wipe out pot.
Heat 1/2 tablespoon oil in a cast-iron skillet on medium-high until hot. Add 3 tomatoes, onions, and garlic; cover and cook, turning vegetables occasionally, until skins are nicely charred, about 10 minutes. Remove from heat. When vegetables are cool enough to handle, remove skins, stems, and cores; discard. Place tomatoes, onions, and garlic in a blender with 1/2 cup reserved broth; blend until smooth. Scrape mixture into saucepan; cook, stirring, over medium heat until thickened and darker, about 6 minutes. Stir in remaining broth, 1/4 teaspoon salt, corn, chickpeas, and chicken. Remove from heat.
Preheat oven to 375 degrees. Place tortilla strips on a rimmed baking sheet; toss with remaining tablespoon oil and 1/8 teaspoon salt. Bake until golden and crisp, about 12 minutes, turning once or twice. Cool.
Finely dice avocado; gently rinse with cold water. Place in a bowl. Finely dice remaining tomato, and add to avocado with lime juice, jalapeno, chopped cilantro, and remaining 1/8 teaspoon salt. Re-warm soup; ladle into bowls. Garnish with avocado mixture, sour cream, and tortilla strips.
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