Put black beans, pork ribs, water, rum, garlic head halves, onion, salt, bay leaves, oregano, cumin, and crushed red pepper in a slow cooker; cover and cook on LOW until the beans are tender and the meat is falling off the rib bones, about 8 hours.
Remove the ribs from the soup. Using a handheld immersion blender, partially puree the soup until it is creamy but still has some texture from whole beans. (Or puree half the soup in a blender and stir it back into the slow cooker.) Pull the meat from the rib bones, shredding it into large pieces, and stir it into the soup.
Mojo: Put the garlic and olive oil in a small microwave-safe bowl and microwave on HIGH until the garlic is golden, 1 to 2 minutes. Stir in the lime juice, coriander, and salt and set aside until ready to serve.
Ladle the soup into warmed bowls and drizzle each serving with some mojo. Pass the remaining mojo.