In a large pot, mix together coconut milk, fresh lime juice, white wine, curry paste, minced garlic, fish sauce, and sugar. Stir well to ensure the sugar and curry paste are fully dissolved.
Bring the mixture to a boil and let it simmer for 2 minutes to enhance the flavors.
Add the mussels to the pot, cover it, and cook while stirring occasionally until the mussels open up, which should take about 5 to 8 minutes.
Once cooked, remove the pot from the heat. Discard any mussels that did not open.
Transfer the mussels and the flavorful broth to a serving dish and sprinkle with fresh cilantro before serving.