In a medium saucepan, combine the coconut milk and granulated sugar over medium heat. Stir until the sugar is completely dissolved (3-5 minutes). Remove from heat and let cool for around 15 minutes.

Stir in freshly squeezed lime juice and optional rum or coconut liqueur if desired. Ensure everything is well combined.

Transfer the mixture to a container, cover, and refrigerate for at least 2 hours to chill thoroughly.

Pour the chilled mixture into an ice cream maker and churn according to the manufacturer's instructions (usually 20-25 minutes). Alternatively, place in a shallow dish and stir every 30 minutes until soft.

Transfer to an airtight container and freeze for about 4 hours until firm enough to scoop.

Let the sorbet sit at room temperature for 5-10 minutes before serving. Serve in decorative bowls, lime halves, or coconut shells.