Start by heating the olive oil in a large pot or Dutch oven over medium-high heat. Season chicken with salt and pepper, then brown for 5 minutes on one side, flip, and brown for 5 minutes on the other, until they are no longer pink inside. Transfer to a plate and set aside.
Top up the oil if needed and add the onion, cooking until the onion begins to soften, about 4-5 minutes. Add the bell pepper, poblano pepper, garlic, and diced green chilies and cook for another 30 seconds.
Add the chicken, chicken broth, diced tomatoes, and red enchilada sauce and stir again. Bring to a boil.
Next, add the array of spices—chili powder, ground cumin, salt, smoked paprika, chipotle chili powder, dried oregano, and a dash of cayenne pepper. Stir these together well to coat the vegetables, allowing the heat to release the spices’ flavors, creating a fragrant base.
Then add the beans, corn, and tortillas, stirring so the tortillas don't stick together, then lower the heat and simmer for 20-25 minutes, until the tortillas have softened.
Remove the chicken from the pot, shred it using two forks, and then return it to the pot. Add the cream cheese and let it melt for 2 minutes. Sprinkle the pepper jack and cheddar cheeses into the soup, stirring well until all the cheese is melted. Remove from the heat and serve with garnishes.