Heat the oil in a large pot over medium-high heat. Add the onion, poblano, garlic, oregano, cumin, and 1/2 teaspoon salt. Cook, stirring often, until the onion is tender, 3 to 4 minutes. Add the tomatoes and cook until they begin to break down, 2 to 3 minutes.
Add the broth, chicken, and hominy. Cover and bring to a boil. Reduce heat; simmer until the chicken is opaque throughout, 3 to 4 minutes.
Top the soup with the Cheddar and cilantro and serve with the lime wedges.