In a blender or food processor, combine the oil, lime juice, garlic cloves, sour orange juice, oregano, thyme, vinegar, salt, and black pepper. Blend until smooth to create the marinade.
Place the pork shoulder in a large bowl or container, and pour the marinade over it, making sure it’s fully coated. Cover tightly with plastic wrap and refrigerate overnight, or for at least 8 hours.
The next day, transfer the pork along with all the marinade into the slow cooker. Cover with the lid and cook on low for 8–10 hours or on high for 5–6 hours, until the pork is tender and falls apart easily.
Once cooked, transfer the pork to a dish and shred with a fork.
Strain ½ cup (120 ml) of the cooking juices and pour it into a small saucepan. Add ½ cup (120 ml) of unsweetened orange juice and 1 teaspoon (3 g) of cornstarch. Stir well and cook over medium heat for a few minutes, until the sauce thickens and becomes glossy.
Serve the pork with the sauce drizzled over the top. If desired, garnish with peeled orange slices added just before serving.