In a large pot, add the lime juice, coconut milk, curry paste, garlic, fish sauce, and sugar.
Cook over medium heat, stirring, for 5 minutes.
Add the mussels and cook in the liquid until the mussels open, stirring occasionally. Add the cilantro about 5 minutes after adding the mussels.
Serve the mussels into individual bowls and add broth to each bowl.