Make the Pork Broth. Add the pork, halved onion, halved garlic bulb, bay leaves, salt, and water to a large pot. The water should cover the pork by about 1 inch.
Bring the broth to a gentle boil, then reduce the heat to maintain a low simmer. As the broth cooks, skim off any foam that rises to the surface. Continue simmering until the pork is very tender and easily pierced with a fork, about 1½ to 2 hours.
Once the pork is tender, remove and discard the onion, garlic, and bay leaves.
Make the Chile Sauce. While the pork is cooking, prepare the chiles. Remove the stems and seeds from chiles and place them in a small saucepan and cover with water. Bring to a gentle simmer and cook for 10–15 minutes, until the chiles are softened.
Transfer the softened chiles to a blender, along with the chopped onion, garlic cloves, dried oregano, cumin, and 1 cup of the reserved chile soaking water. Blend until completely smooth, adding a little more chile water as needed to create a smooth, pourable sauce.
Combine the Posole. Once the pork is tender, remove the aromatics from the broth and pour the chile sauce into the pot with the pork and broth. Stir well to combine.
Add the drained and rinsed hominy and bring the posole to a gentle simmer.
Simmer and Finish. Continue simmering for about 30 minutes, stirring occasionally, allowing the pork, hominy, chile broth to come together. Taste and adjust the seasoning with additional salt as needed.
Serve. Ladle the hot posole into bowls and garnish with plenty of shredded cabbage, sliced radishes and other toppings.