Combine chicken, water, tomatoes, onion, green pepper, beans, lime halves, green chile, garlic, chili powder, and pepper in a large stock pot; bring to a boil. Reduce heat; cover and simmer 2 hours. Remove chicken from broth; cool. Bone chicken and cut into bite-size pieces; return chicken to broth. Refrigerate broth overnight.
Skim off and discard fat that has risen to the surface. Discard lime halves, and stir in salt. Cook soup over medium heat until thoroughly heated. Serve immediately in warm bowls with the following condiments: chopped fresh cilantro, chopped tomatoes, chopped onion, lime slices, shredded Monterey Jack cheese, and fried corn tortillas.