Put oven rack in lower third of oven and preheat oven to 350°F.
Pat pork dry with paper towels, then sprinkle with salt. Heat 1 tablespoon oil in a wide 6-quart heavy pot over moderately high heat until hot but not smoking, then brown pork in 3 batches, turning, about 5 minutes per batch, transferring to a bowl as browned. (Add more oil to pot as needed between batches.)
Pour off all but 1 tablespoon fat from pot. Add onion, bell pepper, and celery and cook over moderate heat, stirring, until softened, 3 to 5 minutes. Stir in garlic, cumin, coriander, turmeric, and cayenne and cook, stirring, 1 minute. Add pork with any juices accumulated in bowl, water, and tomatoes and bring to a simmer.
Cover pot, then transfer to oven and braise pork until very tender, about 1 3/4 hours.
Scatter persimmons over pork and braise in oven, partially covered, 10 minutes more. Stir in scallion greens and salt and pepper to taste.
Cooks' notes: ·Stew can be made, without persimmons or scallions, 1 day ahead and cooled, uncovered, then chilled, covered. Reheat in a preheated 350°F oven, about 30 minutes, before proceeding. ·To peel a tomato, first cut an X in the end opposite the stem and immerse in boiling water 10 seconds. Transfer to ice water, then peel.