Pour the coconut milk into a pan and heat gently. Remove from the heat. Put the sugar and egg yolks in a large bowl and whisk together until pale and the mixture leaves a trail when the whisk is lifted. Slowly add the coconut milk, stirring all the time with a wooden spoon. Strain the mixture into the rinsed-out pan or a double boiler and cook over low heat for 10-15 minutes, stirring all the time, until the mixture thickens enough to coat the back of the spoon. You may find that the mixture starts to separate, and if it does, simply whisk it vigorously until it is smooth again. Do not leat the mixture boil or it will curdle.
Remove the custard from the heat, stir in the dry unsweetened coconut, then let cool for at least 1 hour, stirring from time to time to prevent a skin from forming. Meanwhile, whip the cream until it just holds its shape. Keep in the refrigerator until ready to use. When the custard is cold, add the coconut rum, if using, and mix well together.
If using an ice-cream machine, fold the cold custard into the whipped cream, then churn the mixture in the machine following the manufacturer's instructions. Alternatively, freeze the custard in a freezerproof container, uncovered, for 1-2 hours, or until it starts to set around the edges. Turn the custard into a bowl and stir with a fork or beat in a food processor until smooth. Fold in the whipped cream. Return to the freezer and freeze for an additional 2-3 hours, or until firm or required. Cover the container with a lid for storing.
To prepare the tropical fruits, thinly slice and put in a large shallow dish. Sprinkle sugar and coconut rum, if using, over the fruit, then cover and let chill in the refrigerator for 2-3 hours before serving with the ice cream.