BRING coconut milk, milk and ginger to simmer in medium saucepan. Remove from heat. Cool slightly.
BEAT egg yolks and sugar in medium heatproof bowl with wire whisk until fine yellow ribbons form. Gradually add 1/2 cup of the coconut milk mixture, whisking until well blended. Place bowl over a double boiler. Whisk in remaining coconut milk mixture. Cook until mixture thickly coats the back of a wooden spoon, stirring constantly. Remove from heat. Cool completely in refrigerator or place bowl over an ice bath. When cool, strain through a fine sieve. Stir in cream and rum until well blended.
POUR into an ice cream maker. Freeze according to manufacturer's directions.