To make the ice cream, in a large, heavy saucepan, combine the cream, coconut milk, whole milk, rum, and sugar. Bring to a simmer over medium heat, stirring to dissolve the sugar. Remove from the heat.
In a medium bowl, beat the eggs until pale yellow and frothy. Add about 1/2 cup of the hot cream mixture, and whisk to combine. Add the egg mixture to the saucepan with the remaining hot cream and whisk. Return to medium heat and cook, stirring constantly with a heavy wooden spoon until thick enough to coat the back of a spoon, 4 to 5 minutes. Remove from the heat and strain through a fine mesh strainer into a clean container. Refrigerate until well chilled, at least 3 hours. Pour into an ice cream maker and process according to the manufacturer's instructions. Transfer to a plastic container and freeze until ready to serve.
Yield: 1 quart