In a medium (2 to 3-quart) saucepan, combine coconut cream, coconut water, agave nectar, and sugar. Bring up to a bare simmer over medium heat, stirring often.

Transfer the mixture to a blender, cover tightly, and blend at high speed for 30 seconds. Add the vanilla bean paste, rum, and salt, and blend for another few seconds, just to combine.

Chill the mixture for at least four hours, or overnight.

Churn the mixture according to your ice cream maker’s instructions. Select the ice cream function for a slightly more aerated final product, or gelato for a denser texture. Chill for at least two hours before serving for a scoopable texture.