Remove the meat from the chicken and shred it into 1-inch pieces; discard the skin and bones. Set aside about 1 1/2 cups of the meat for the soup; reserve the remaining chicken for another use.
In a large Dutch oven or saucepan set over medium heat, heat the oil. Add the onion and cook until translucent and fragrant, about 8 minutes. Add the garlic and cook until softened but not browned, 3 to 4 more minutes. Add the hominy with the liquid from the cans, tomatoes, salsa verde and green chiles and cook until the mixture just comes to a boil, about 2 minutes. Add the chicken broth, coriander, cumin and oregano. Stir, increase the heat and return to a boil. Reduce the heat to medium and simmer to allow flavors to develop, at least 10 minutes. Season with salt and pepper. Add the shredded chicken and cook until heated through, about 3 minutes.
To serve, divide the Pozole among 4 bowls and garnish as desired.
You can use the remaining half can of diced tomatoes in Crustless Quiche.