Measure 7 quarts of water into a kettle or stockpot

Add the pork shoulder, the garlic, and the hominy to the water

Bring the stock to a boil and then turn down to a simmer

Tear the chiles into large, flat pieces and toast them, a few at a time, on a griddle or heavy skillet set over medium heat, using a metal spatula to press them firmly against the hot surface until they crackle and blister, then flipping them over and pressing them down again. Remove to a bowl, cover with boiling water, weight with a place to keep them submerged, and soak for 30 minutes.

Drain, place in a blender jar, and add 1/2 cup water; blend until smooth. Strain through a medium-mesh sieve into the simmering soup, then mix well.

Generously season the soup with salt and let simmer for another hour or so.

Remove the pork shoulder from the soup and let cool.

When cool enough to handle, remove any fat, and chop the meat into 1 inch pieces and roughly shred it.

Just before serving, add the meat to the pot and let simmer a few minutes to reheat

Ladle the soup into large bowls, top each one with a portion of shredded cabbage and some sliced radishes

Add onion, oregano, and lime wedges as desired

The tortilla chips are a crunchy accompaniment to enjoy between big spoonfuls of the soup.