Prepare dry chili peppers. Remove stems, veins, and seeds. Cut them into small pieces, and quickly toast them on a hot, dry skillet, stirring. (For more portions of this soup, roast dry chiles in hot 350F for 5 minutes.) Place toasted chili peppers in a bowl of a blender, and pour about a cup of hot water over. Make them submerge by pressing down. Leave for 20-25 minutes to reconstitute.
Dice onion and garlic.
Cut pork into ~2-inch cubes. Heat oil (or pork fat) in a heavy skillet over high heat. Cook every batch separately for about 1 minute, constantly stirring, until pork surface becomes white.
Add onions and garlic and continue cooking for another 2 minutes, stirring. Add water or stock, oregano, cumin, season with salt and pepper. Bring to boiling, reduce the heat to low and let simmer for about 20 minutes.
Blend chili peppers with their water and add the puree to the soup. Stir and continue simmering for another 10 minutes or until pork is fork-tender.
Prepare soup toppings. Thinly slice radishes, jalapeno, scallions, lime or key lime, chiffonade cabbage leaves, and pick cilantro leaves.
When pork is tender, add canned posole (hominy corn kernels) and bring the soup to final boiling. Serve toppings separately, buffet style, to be stirred into the bowl of hot soup individually.