If using dried hominy, place it in a large soup pot. Add water to the pot to cover the hominy by at least 3-inches. Add the head of garlic. Don’t add salt now or the hominy will toughen. Bring to a boil and simmer over medium-low heat, partially covered, for 4 to 5 hours, until hominy is tender and has begun to “bloom” or open up. Occasionally skim the foam from the top as the hominy cooks and make sure it doesn’t dry as it cooks, adding more hot water if need be. If using canned or pre-cooked hominy, start with step below.
Meanwhile, place the chicken (and pork, if using), in a large soup pot. Add water to cover the top layer of chicken by at least 2 inches. Add the onion, cilantro and the tablespoon of salt and bring to a boil. Simmer, partially covered, until chicken is cooked through and tender, about 35 minutes. Drain, reserving the cooking broth. When the chicken is cool enough to handle, remove the skin and bones and shred the meat into bite-sized pieces.
In the soup pot, combine the cooked hominy and its broth (discard the garlic head), or the canned hominy and 2 cups water, with the shredded chicken and its broth. Taste for salt, add more if need be, and simmer all together for 10 minutes more.