Place oven rack in lower third of oven and preheat oven to 350˚F.
Pat pork dry with paper towels, then sprinkle with salt. In a 6-quart Dutch oven or other large, heavy, oven-safe pot, heat the olive oil over medium-high heat until hot but not smoking. In 3 batches, brown the pork in the pot, about 2 1/2 minutes per side. Transfer to a large bowl as you go. Add more oil if needed between batches.
Pour off all but 1 tablespoon of fat from the pot. Over medium heat, cook the onion, bell pepper, and celery until softened, 3-5 minutes. Stir in the garlic, cumin, coriander, turmeric, and cayenne, and cook, stirring, for 1 minute. Transfer the pork and any accumulated juices from the bowl back into the pot. Add the water and tomatoes, and bring to a simmer over medium heat.
Cover the pot, then transfer to the oven. Braise about 1 hour 45 minutes, until very tender. Add the persimmons to the pork, partially cover the pot, and cook another 10 minutes. Remove from oven, add salt and pepper to taste, and top with the chopped scallions.