1. Cut the pork shoulder into large pieces and bring to the boil in a large pot with the water, the halved onion, 4 garlic cloves and salt; skim off the foam and simmer gently for 1½ hours.
2. Meanwhile, stem and deseed the guajillo and ancho chilies, toast them in a dry pan for a few seconds, then soak in hot water for 20 minutes.
3. Blend the soaked chilies with the remaining garlic and a little soaking liquid until smooth and pass through a sieve.
4. Add the drained (if canned, rinsed) hominy and the chili purée to the soup, sprinkle in the Mexican oregano and cook for another 30-40 minutes until the corn kernels "bloom" and the meat is tender; remove the onion and pull the meat into large pieces with a fork.
5. Meanwhile, shred the cabbage hair-thin, slice the radishes, finely chop the red onion and cut the limes into wedges.
6. Ladle the soup into deep bowls and let everyone add cabbage, radish, onion, another pinch of oregano and a squeeze of lime; serve with tostadas.
Tip: Pozole is a festive dish – in Mexico almost every household makes it on 15 September, Independence Day.
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