Place coconut milk and coconut cream in the refrigerator.
Place water and rice in a blender. Blend until rice is coarsely ground. Pulse until rice is finely ground. Pour mixture into a jar or bowl. Add vanilla bean halves and cinnamon stick pieces. Cover and refrigerate, 8 hours to overnight.
Strain rice 'milk' mixture through cheesecloth. Scrape vanilla seeds from the pod and add to the milk. Discard rice pulp and cinnamon stick pieces.
Blend strained rice milk, coconut milk, coconut cream, brown rice syrup, and rum until well combined.
Pour into an ice cream maker and freeze according to manufacturer's directions, 15 to 20 minutes. Freeze the mixture for 2 to 3 hours before serving.
