Cut the pork shoulder into large cubes and rub with salt, pepper, cumin, smoked paprika, and dried oregano.
In a large Dutch oven or pot, heat olive oil or lard over medium-high heat. Sear the pork cubes on all sides until golden brown.
Add chopped white onion, garlic cloves, bay leaves, orange juice, and lime juice. Pour in enough water to just cover the meat.
Cover and simmer over low heat for 2.5-3 hours, or until the pork is very tender and easily shredded.
Once the pork is tender, remove it from the liquid and shred it with two forks. In a skillet or under a broiler, crisp up the shredded pork until caramelized and slightly crispy.
Lightly warm the corn tortillas in a dry skillet or microwave to keep them pliable.
Serve the carnitas on the warm tortillas, topped with chopped red onion and fresh cilantro. Offer lime wedges for squeezing fresh lime juice on top.
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