Trim excess fat from the pork shoulder and rub it thoroughly with sea salt, black pepper, ground cumin, and dried oregano.
Place the seasoned pork into a slow cooker along with the minced garlic, fresh orange juice, and lime juice.
Cover and cook on low for 7 to 8 hours (or high for 4 to 5 hours) until the meat is tender enough to shred easily with a fork.
Remove the pork from the slow cooker and shred it into bite-sized pieces using two forks, discarding any large remaining bits of fat.
Heat the extra virgin olive oil in a large skillet over medium-high heat.
Add the shredded pork to the skillet and press down with a spatula, cooking for 3 to 4 minutes until the edges become crispy and golden brown.
Warm the corn tortillas in a dry skillet or over a low open flame until they are soft and pliable.
Divide the crispy carnitas evenly between the tortillas and garnish with diced red onion, chopped cilantro, and thinly sliced radishes.