When the pork is done cooking, remove and place onto a cutting board. You're going to find some large pieces of fat because pork shoulder has a lot of it. Find these pieces and remove what you can find.
Shred the remaining pork using your finger or a fork if too hot to handle. Add back to the pot.
Drain the cans of hominy and add the hominy to the pot.
Add the reserved pureed peppers into the pot and stir.
Add the ¼ teaspoon of dried oregano, stir and simmer for 5 minutes to let the flavors meld together.
Skim off any fat that may be floating along the top but if you are not serving until the following day, you'll be able to easily skim off a layer of solidified fat with a spoon after being refrigerated over night.
I also think the posole tastes better a day after you make it.