1. Remove the meat from the chicken and shred it into 1-inch pieces; discard the skin and bones. Set aside about 1½ cups of the meat for the soup; reserve the remaining chicken for another use.
2. In a large Dutch oven or saucepot set over medium heat, heat the oil. Add the onion and cook until translucent and fragrant, about 8 minutes. Add the garlic and cook for 3 to 4 more minutes. Add the oregano, coriander and cumin, and cook until slightly darkened and fragrant, 2 to 3 minutes. Add the green chiles, chopped tomatoes, salsa verde and cook until it just comes to a boil, about 2 minutes. Add the hominy and chicken broth and return to a boil. Reduce the heat to medium and simmer for at least 10 minutes to allow flavors to develop. Season with salt and pepper. Add the shredded chicken and cook until heated through, 3 minutes.
3. To serve, divide among 4 bowls and garnish as desired.