Heat oil in a large pot over medium heat. Add pork, chicken, Hatch chile peppers, onion, jalapeno peppers, and garlic. Saute, turning meat, until lightly brown, about 8 minutes.

Pour 2 cups stock into pot; add enough water to cover meat by 3/4 inch. Bring to a boil. Reduce heat and simmer, covered, for 45 minutes. Remove pork and chicken and place on a plate; let cool until safe to handle. Remove garlic cloves, mash, and add back to soup.

Shred pork and chicken into small chunks. Add salsa to the pot. Let soup simmer for 15 to 30 minutes more.

Place hominy, shredded meat, remaining 4 cups stock, New Mexico chile powder, coriander, salt, and pepper into the pot of soup. Bring to a boil; reduce heat and simmer, uncovered, at least 30 minutes.

Serve with cabbage, yogurt, cilantro, and radishes to garnish.