Preheat the oven to 350 degrees F (175 degrees C).

Put sugar in a heavy-duty saucepan and cook over medium-low heat, stirring with a wooden spoon, until melted and golden, about 10 minutes. Pour hot caramel into a baking dish and swirl to coat the bottom and sides. Let it cool.

Blend eggs, cream cheese, and sweetened condensed milk. Transfer to a bowl, add coconut milk, cream of coconut, evaporated milk, rum, and vanilla extract. Whisk until combined.

Pour the mixture into the caramel-coated dish. Place the dish in a water bath inside a roasting pan, bake for 1 hour until set. Cool on a wire rack.