Place cooked rice in a saucepan with 1 cup of water.

Bring to a simmer, cover, and cook for 15 minutes.

Set aside, covered, for 15 minutes longer to allow the rice to fully absorb the water.

In a separate heavy saucepan, combine coconut milk, 1/4 cup of heavy cream, and the split vanilla bean.

Scald the mixture (heat until just before boiling), then remove from heat.

Remove the vanilla bean and scrape the seeds back into the coconut milk mixture; discard the pod.

In a mixing bowl, beat egg yolks with 1/3 cup of sugar until light and pale.

Gradually add the scalded coconut milk mixture to the egg yolks, whisking constantly to temper the eggs.

Return the custard mixture to the saucepan.

Cook over medium-low heat, stirring constantly with a wooden spoon, until the custard thickens and coats the back of the spoon (nappe).

Be careful not to overcook the custard, as it can curdle.

Remove from the heat and continue stirring for 1 minute to prevent further cooking.

Strain the custard through a fine-mesh sieve into a bowl to remove any lumps.

Stir in salt and 2 tablespoons of white rum.

Gently fold the cooked rice into the custard, ensuring it is evenly distributed.

Cover the bowl with plastic wrap, pressing it directly onto the surface of the pudding to prevent a skin from forming.

Chill in the refrigerator for at least 4 hours, or preferably overnight, to allow the flavors to meld and the pudding to fully set.

Whip the remaining heavy cream until stiff peaks form.

Fold the whipped cream into the chilled rice and custard mixture, being careful not to deflate the cream.

Spread the pudding evenly in an oven-proof dish, about 9 inches square, suitable for serving.

Refrigerate until just before serving.

In a separate bowl, mix the sliced tropical fruits with the remaining white rum.

Preheat the broiler.

Sprinkle the remaining sugar evenly over the surface of the pudding.

Place the dish under the preheated broiler until the sugar on top is lightly browned and caramelized, watching carefully to prevent burning.

Serve immediately with the rum-soaked tropical fruit on the side.