In a 6- to 8-quart pan, combine pork with 1/3 cup water. Cover and cook over medium-high heat, stirring occasionally, until meat is very juicy, 15 to 20 minutes. Uncover, increase heat to high, and cook, stirring often, until liquid is evaporated and meat is browned, 20 to 30 minutes. Lift out meat and set aside.
Reduce heat to medium. If you have leftover pork fat in the pan, discard all but 3 tablespoons; if not, add oil. Add onion, garlic, and cumin; stir and cover. Cook, stirring occasionally, until onions are soft, about 8 minutes.
Return meat and any accumulated juices to pan. Add tomatoes and broth. Break up tomatoes with a spoon. Bring to a simmer, then reduce heat, cover, and simmer, 60 minutes.
Add chiles and oregano to pan. Cover and cook until pork is very tender when pierced and flavors are blended, about 15 minutes. Season to taste with salt and pepper. Serve topped with cilantro leaves. Garnish with lime to squeeze over chile.