Peel the garlic cloves and reserve 2 for the chile sauce. Slice the remaining garlic. In a 7 to 8 quart heavy kettle bring water and stock just to a boil with sliced garlic and pork. Skim fat from the surface and add oregano. Gently simmer pork, uncovered, until tender, about 1 1/2 hours.
You might want to wear protective gloves for this step. Discard stems from chiles, and in a bowl, combine chiles with boiling-hot water. Soak chiles, turning them occasionally, for 30 minutes. Cut onion into large pieces and place in a blender or food processor with chiles and soaking liquid, reserved 2 cloves of garlic, and 2 teaspoons salt until blend until smooth.
Transfer pork to a cutting board and reserve stock mixture. Using 2 forks, shred the pork. Discard the pork bones. Rinse and drain hominy. Return pork to stock mixture and add chile sauce, hominy, and remaining teaspoon salt. Simmer posole 30 minutes and, if necessary, season with salt and pepper as desired. Posole may be made 2 days ahead and chilled, covered.
While posole is simmering, prepare tortillas. Stack tortillas and cut them in half and then cut the halves crosswise into thin strips. In a 9 to 10-inch skillet heat 1/2 inch oil until hot but not smoking and fry tortilla strips in 3 or 4 batches, stirring occasionally, until golden, 1 to 2 minutes. Transfer tortilla strips with a slotted spoon as fried to brown paper or paper towels to drain. Transfer tortilla strips to a bowl. Tortilla strips may be made 1 day ahead and kept, covered, at room temperature.
Serve posole with tortilla strips and bowls of accompaniments.