Measure 1 cup of the drained beans into a bowl with ½ cup of the measured broth. Mash until mostly smooth and set aside.
Heat the oil in a 5 to 6 quart heavy pot over medium heat. Cook the onion and poblano for 6 to 8 minutes until softened with a few golden edges. Stir in the garlic, cumin, coriander, oregano, salt, and pepper for 30 to 45 seconds until fragrant.
Add the remaining 3½ cups broth, mashed bean mixture, whole beans, green chiles with their liquid, and corn. Stir until the bean mash disperses evenly.
Bring to a gentle boil, then reduce to medium-low and simmer uncovered for 10 minutes, stirring twice.
Stir in the shredded chicken and simmer for 4 to 6 minutes until the mixture reaches 165°F throughout. Check consistency — broth should coat a spoon lightly. Add broth by the tablespoon if too thick.
Remove from heat, stir in the lime juice, and taste before adding more salt or citrus. Divide among six bowls and add 2 tablespoons sour cream to each bowl, then add toppings as desired.