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 broth just to a boil with sliced garlic and pork. Skim the surface and add oregano. Gently simmer pork, uncovered, until tender, about 1 1/2 hours.
While pork is simmering, wearing protective gloves, 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 in a blender puree with chiles and soaking liquid, reserved 2 cloves of garlic, and 2 teaspoons salt until smooth.
Transfer pork with tongs to a cutting board and reserve broth mixture. Using 2 forks, shred the pork. Discard the pork bones. Rinse and drain hominy. Return pork to broth mixture and add chile sauce, hominy, and remaining teaspoon salt. Simmer posole 30 minutes and, if necessary, season with salt. Posole may be made 2 days ahead and chilled, covered.
While posole is simmering, stack tortillas and halve. Cut 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.
Recipe courtesy of Gourmet Magazine