Put the pork, half the onion, 3 of the garlic cloves, the bay leaves and 1 teaspoon of the salt in a large pot with the water. Bring to a bare simmer, skimming off the grey foam that rises in the first ten minutes.
Cover partly and simmer gently for about 2 hours, until the pork is tender enough to pull apart with a fork. Keep it at a lazy bubble, never a rolling boil.
While it cooks, toast the stemmed, seeded chiles in a dry pan for 5 to 10 seconds a side, just until fragrant. Cover them with hot water from the pot and soak 20 minutes until soft.
Blend the soaked chiles with the remaining onion and garlic and a cup of the cooking broth until completely smooth.
Strain the chile puree through a fine sieve directly into the pot, pressing the solids and discarding the skins left behind. This step is what makes the broth smooth rather than gritty.
Add the hominy, the oregano crushed between your palms, and the remaining salt. Simmer uncovered 30 minutes more so the flavours marry and the hominy takes on the broth.
Lift out the pork, shred it, return it to the pot, and taste for salt. Serve with the garnishes alongside.