Coconut milk adds a soft, nutty taste to this Asian fish chowder, while also thickening it. When cooking with coconut milk or cream, don't let the liquid boil or it may curdle or separate.
Heat butter in a large saucepan or small stockpot over medium heat. Add onion, and cook, stirring occasionally, until translucent, 3 to 4 minutes. Add garlic and lime leaves; cook 1 minute. Add fish stock, potatoes, celery, green beans, and ginger; bring to a simmer over medium heat. Cook 7 to 9 minutes.
Reduce heat to low. Add coconut milk, and bring almost to a simmer. Add fish; without stirring, simmer until the fish is opaque and vegetables are tender, about 5 minutes. Season with salt and pepper. Ladle the soup into bowls; garnish with chives and coconut. Serve immediately.