Combine the coconut milk, moss and ice cubes in a blender and blend well until all of the Irish moss pieces have been pulverized. Stop blending halfway through and scrape down the sides and lid of the blender with a spatula to get any remaining pieces of moss. The warmer the mixture becomes the thicker and pastier it will be, and it will begin to stick to the sides of the blender, so try to keep it cool. The coconut milk can be put in the refrigerator to stay cold before blending.
For the drink:
Heat 2 cups water in a saucepan over medium heat with the milk, 1/3 cup of the Irish moss mixture, the linseed, isinglass and gum arabic. Bring to a boil and allow it to simmer until all of the ingredients have been dissolved and incorporated and the mixture thickens, about 30 minutes. Turn down to low heat and add the sugar, vanilla and nutmeg and continue cooking, stirring occasionally for another 10 minutes. Remove from the heat. Any tiny pieces of moss remaining in the mixture are edible, but can be strained out at this point if preferred.
Let cool in the fridge for at least 3 hours, as the beverage is traditionally served cold. Stir in the rum. The mixture will thicken more as it cools off. Depending on your personal taste, you may loosen the mixture slightly with water, milk or both and adjust the sweetness before serving.