Heat a large pot over medium heat. Add the diced onion and cook for 3 minutes until it softens.
Stir in the minced garlic, cumin, oregano, and chili powder. Cook for 1 minute until the spices smell fragrant.
Pour in the chicken broth and bring the mixture to a simmer. Add the chicken breasts and cook for 15 minutes until they reach an internal temperature of 165°F.
Remove the chicken from the pot and shred it with two forks. Return the shredded chicken to the broth.
Add the drained hominy and simmer for 10 more minutes. Season with salt and pepper to taste.
Squeeze the juice of one lime into the soup and stir. Cut the second lime into wedges for serving.
Ladle the soup into bowls and top with chopped cilantro and sliced radishes. Serve with lime wedges on the side.