Preheat the oven to 350.
In a Dutch oven or oven proof skillet, mix together the tomato paste, salt, pepper, garlic cloves, and pork cubes, then pour the broth over all and stir some more. Cover and put in the oven for 1 1/2 to 2 hours until the pork is very tender.
Cool and refrigerate overnight.
Skim any solidified fat off. Let the pork sit out for 30 minutes to get up to room temperature, then cook in a large non-stick skillet over medium heat until most of the liquid is evaporated, 10-15 minutes.
Remove from the heat and stir in the lime juice.