Heat the oil in a large saucepan over medium heat.
Add the onion and ¼ teaspoon each salt and pepper and cook, stirring occasionally, until soft and beginning to brown, 10 to 12 minutes.
Add the broth, tomatoes, and chili and bring to a boil.
Stir in the chicken and hominy and simmer until heated through, 3 to 4 minutes.
Serve the posole with the lime wedges.