Measure seven quarts of water into a stockpot and add the pork shoulder and garlic, simmer for about one hour.
Add in rinsed hominy and simmer, uncovered, for another 30 minutes.
Meanwhile, tear the chilies in half, toast in a large cast-iron pan, pressing down on them all until they blacken and blister in places — about 30 seconds to one minute per side.
Set in a bowl, cover with hot water and let soak for 30 minutes.
Puree the chilies along with 1 1/2 cup of the soaking liquid and oregano.
Strain into the soup, mix well. Season generously with salt.
Let simmer another hour.
Simmer until the pork is tender, then remove all of the pork pieces.
Cool the meat, remove meat from bones, discard cartilage and bones. Shred the pork shoulder, reserve all of the meat.
Re-season the soup with salt and lime juice.