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Heat oil in a large Dutch oven over high heat. Season pork generously with salt; then arrange in Dutch oven. Cook 10 minutes, until browned on all sides. Add onion, garlic, half of lime juice, chili powder, oregano and cumin. Pour in chicken broth and bring to a boil. Reduce heat to medium-low, cover, and continue to simmer until pork is very tender, about 2 1/2 hours.
Preheat oven to 400 degrees.
Combine shredded cabbage, carrots, cilantro and onion in a bowl with rest of lime juice and a sprinkle of sea salt; toss to combine. Let sit to marinate.
Transfer pork to a large baking sheet, reserving cooking liquid. Drizzle with a small amount of reserved cooking liquid and lightly season with sea salt.
Bake in preheated oven until browned, about 30 minutes. Drizzle more cooking liquid on meat every 10 minutes, and use two forks to shred meat as it browns.
Place tortillas on a large baking sheet and warm in the oven. Flip and repeat on other side. Wrap tortillas in a clean dish towel to keep warm.
Add shredded pork to each tortilla; top with cabbage carrot slaw, more cilantro and your favorite taco toppings.