Rinse the bulgur wheat under cold running water, then cook in a large pan of boiling salted water according to the packet instructions. Drain and leave to cool completely.
Meanwhile, toast the pistachios in a dry frying pan over a medium heat, then roughly chop and add to a large bowl. Pick, finely chop and add the mint and parsley leaves, discarding the stalks. Trim the radishes and spring onions, then roughly chop along with the tomatoes. Add all the chopped veg to the bowl with the lemon juice, 4 tablespoons of olive oil and a good pinch of salt and pepper. Toss well.
Fluff up the bulgur wheat with a fork, then add to the bowl and mix well. Halve the pomegranate and bash it cut-side down with a wooden spoon so the seeds come tumbling out into the bowl. Season to taste, sprinkle over the sumac and tuck in.