Place the curry paste and ginger in a medium pot. Whisk in the broth, coconut milk, and 1 teaspoon salt and bring to a boil.
Add the mushrooms, green beans, and carrots and simmer until just tender, 3 to 5 minutes.
Add the tofu and snow peas and simmer until the snow peas are bright green, about 1 minute more.
Stir in the lime juice. Sprinkle with the basil and serve with the chili garlic sauce.