Shred the papaya: peel, halve, and seed the green papaya, then shred into long thin strands (4 cups). Soak in ice water for 10 minutes, drain well, and pat dry.
Pound the base: pound the chiles and garlic with a pinch of salt into a rough paste in a large mortar. Add the palm sugar and dried shrimp; pound until the sugar dissolves and the shrimp breaks down.
Make the dressing: add the lime juice, fish sauce, and tamarind water; stir with the pestle until dissolved. Taste - it should be assertively sour, salty, and hot with a background sweetness.
Bruise the beans and papaya: add the long beans and pound a few times until bruised. Add the papaya shreds in two batches, pounding and folding until lightly bruised but still crunchy.
Finish and serve: fold in the cherry tomatoes and half the peanuts. Plate up, spoon over remaining mortar dressing, scatter with the rest of the peanuts, and serve immediately with lime wedges.