Rub the pork shoulder thoroughly with sea salt, black pepper, dried oregano, ground cumin, and minced garlic.
Place the seasoned pork into a slow cooker and pour in the water.
Cover and cook on low for 8 hours or on high for 4 hours until the meat is tender and shreds easily with a fork.
Transfer the pork to a cutting board and shred it completely, discarding any large pieces of rendered fat.
Heat the avocado oil in a large cast-iron skillet over medium-high heat until shimmering.
Spread the shredded pork in an even layer across the skillet and press down firmly with a spatula.
Sear the meat for 3 to 5 minutes without stirring until the bottom becomes deeply browned and crispy.
Warm the corn tortillas in a separate dry pan over medium heat for 30 seconds per side until soft and pliable.
Divide the crispy pork evenly between the warm tortillas.
Top the tacos with diced red onion, fresh cilantro, and sliced radishes, then serve with a fresh lime wedge.