Peel the lychees, and remove the flesh from the seeds (they will break into 3 or 4 pieces, which is fine). Set aside.
Bring a small pot of water to a boil. Add the beans, and cook until they are bright green and crisp-tender, about 4 minutes. Strain, and set aside to cool.
Meanwhile, pulse the tomatoes, chopped shallots, ginger, 1 tablespoon of the oil, sugar, curry, cayenne and 1 teaspoon salt in a food processor until it becomes a puree.
Heat the remaining 2 tablespoons oil in a medium saucepan over medium heat. Add the sliced shallots, and cook, stirring frequently, until a deep golden brown, about 5 minutes. Add the tomato puree, and cook, stirring and scraping up the bottom frequently, until the paste is thick, dry and a few shades darker, 15 to 20 minutes. Add water, a tablespoon at a time, if the paste sticks to the bottom of the pot. (Be sure to scrape the side of the pot down, as some cooked paste may cling to it, and you want to make sure all the paste makes it into the sauce there's flavor in there!)
Add 2 cups of water and the coconut milk to the pot, and bring to a gentle simmer. Add the eggplant, and simmer until soft, about 6 minutes. Add the peppers, and cook until soft, about 6 minutes. (The sauce should be a bit looser than marinara; continue to gently simmer if necessary.) Add the shrimp and green beans, and cook, stirring occasionally, until they are pink and cooked through, 3 to 4 minutes. Stir in most of the basil and cilantro, and season with salt.
Transfer the curry to a serving bowl. Top with the lychees and the remaining basil and cilantro. Serve with rice and lime wedges.
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