Combine coconut milk, coconut cream, sugar, vanilla, salt and white rum. Mix well to dissolve sugar and incorporate all ingredients well.
Pour cream into a frozen loaf pan and freeze overnight until properly set.
The next day, cut pineapple into 1 inch pieces. Cook pineapples in a small saucepan over high heat with sugar and water until the water reduces and becomes syrupy, about 5 minutes.
Blend half of the pineapple mixture and add it to the other half of the pineapple pieces in the saucepan. Cook on low heat until thick, about 2 mins. Set aside until fully cooled.
Remove frozen cream from the freezer. Cut through the cream with a sharp knife and remove from the loaf pan with a spoon. Add all frozen pieces to the food processor along with the reserved ¼ cup of coconut cream. Blend until smooth but not runny, about 1 minute.
Add half of the blended cream back to the loaf pan. Spoon in half of the pineapple mixture and use the tip of a knife to swirl the coulis into the cream. Top with the remaining half of cream. Spoon the remaining coulis on top and swirl with the tip of a knife.
Set the ice cream back into the freezer to set for at least 4 hours. Remove from the fridge about 10 minutes before serving so the ice cream is easier to scoop. Enjoy!