Combine pineapple, brown sugar, and lime juice in a medium saucepan over medium heat. Cook, stirring occasionally, until the pineapple breaks down and the mixture thickens to a jam-like consistency, 15-18 minutes. The liquid should reduce by half and coat the back of a spoon. Let cool completely, then refrigerate until cold.

Open the chilled coconut milk cans and scoop the thick cream from the top into a large bowl, leaving the thin liquid behind for another use. Beat with an electric mixer on medium-high speed until fluffy and doubled in volume, about 3 minutes.

Add heavy cream, sugar, rum if using, vanilla, and salt. Beat on medium speed until the sugar dissolves and the mixture is smooth and thick, 2-3 minutes. Do not overbeat or the cream may separate.

Pour the base into your ice cream maker and churn according to manufacturer's instructions until it reaches soft-serve consistency, typically 20-25 minutes.

Spread one-third of the ice cream into a loaf pan or freezer container. Dollop half the cooled pineapple swirl on top and use a knife to swirl it through. Repeat with another third of ice cream and remaining pineapple. Top with final third of ice cream. Drag a knife through once more for marbling.

Cover tightly with plastic wrap pressed directly onto the surface, then foil. Freeze until firm enough to scoop, at least 4 hours or overnight. Let sit at room temperature for 5 minutes before serving for the creamiest texture.