Bring the water and clam juice to a boil over high heat in a large soup pot with the scallion whites, lemongrass, ginger, garlic, and chili paste. Cover and adjust the heat to maintain a simmer, and cook for 10 minutes. Add the fish sauce.
Slice the fish into large chunks and add to the broth with the mushrooms and carrots. Cook over medium heat, uncovered, until the fish is opaque and cooked through, about 4 minutes. (Remove and discard lemongrass.) Stir in the cilantro and scallion greens and season with salt. Spoon into warm bowls and serve with lime wedges.
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