In a large soup pot, warm the olive oil over medium heat. Add onion, poblano pepper, and jalapeno pepper. Season with salt and black pepper. Cook until softened and fragrant. Add the taco seasoning and oregano. Stir again to combine. Add the canned green chiles, salsa verde, enchilada sauce, and chicken broth. Bring to a boil and reduce to a simmer. Cook for about 10 minutes.
Add the cream cheese and whisk until fully incorporated. Stir in the chicken and the heavy cream. Warm through for about 5 minutes. Adjust salt and pepper to taste and add the lime juice and cilantro.
Serve the soup by placing a 1/2 cup of rice in a bowl. Spoon the soup on top, and top with your favorite garnishes. I used green onion, sour cream, cilantro, avocado, and chips. Enjoy!