Cut pineapple into chunks (I use a pineapple corer then cut into chunks).
In a blender, add the pineapple, banana, sugar, and lime juice. Blend everything until very smooth. I did not have to do it in batches, but you may need to depend on the size of your blender or the amount of pineapple. Adjust the amount of sugar if needed. Strain the mixture through a fine-mesh sieve into a large bowl. Chill in the refrigerator for at least 3-4 hours before churning in your ice cream maker.
Before processing in your ice cream maker, stir in the Cointreau.
Transfer the chilled mixture into the chilled container of your ice cream maker and process according to the manufacture’s instructions. Chill until frozen and serve.