Place pork shoulder, onion halves, garlic, bay leaf, and cloves in a large pot. Pour in 14 cups water (pork should be submerged). Bring to a simmer and cook, skimming as needed and turning pork occasionally, until meat is cooked through and tender but not yet falling apart, 2 1/2–3 hours. Transfer pork to a plate; let cool slightly. Strain broth into a large bowl, then transfer back to pot. Slice pork into 1/2" slices and add to broth.
Meanwhile, preheat oven to 350°F. Toast all the chiles on a rimmed baking sheet until brown (do not char) and starting to lightly puff in places, about 5 minutes. Bring 3 cups water to a boil in a medium saucepan; add chiles. Remove from heat and let chiles soak until softened, 10–12 minutes. Blend chiles and cooking liquid in a blender until smooth; season with salt.
Add chile purée to pork and broth. Bring to a simmer, add hominy, and cook, skimming off fat from surface, until pork is so tender it’s nearly falling apart, 45–60 minutes; season with more salt.
Divide posole among bowls, top with onion, cabbage, radishes, and oregano. Serve with lime wedges.