Combine the coconut milk, cream, milk, sugar and vanilla extract in a heavy saucepan
over medium heat. Cook until the milk is hot but not boiling and the sugar is dissolved,
about 10 minutes. Remove from the heat.
Place the egg yolks in a bowl and whisk to mix. Whisking constantly, slowly pour
in 1 cup of the hot milk mixture and whisk until smooth.
Slowly pour the egg mixture
into the saucepan, whisking constantly until well combined.
Place the saucepan over
medium heat and stir the mixture constantly until it is thick enough to coat the
back of a spoon, 6 to 8 minutes. The mixture should never boil.
Strain the mixture into a bowl and stir in the rum and coconut. Cool to room
Freeze in an ice cream maker according to the manufacturer's instructions.