Add the sugar to the metal pan and melt over the stove at low-medium heat moving it with a wooden spoon until golden color. Turn the heat off a little before the desired color as it will continue to darken and it will burn if you are not careful. Very carefully and using oven mitts, swirl pan around while sugar is still soft to cover bottom and sides. It will get thicker as it cools and eventually harden as a lollipop. Set pan aside to cool.
Combine the eggs, cream cheese, and sweetened condensed milk in a blender. Cover, and blend on High until well combined; transfer to a large bowl. Pour the cream of coconut, coconut milk, evaporated milk, coconut-flavored rum, and vanilla extract into the blender; cover and blend on High until well combined. Pour into the bowl with the cream cheese mixture. Whisk until incorporated. Pour the flan mixture into the caramel-coated baking dish.
Line a roasting pan with a damp kitchen towel. Place the filled baking dish on the towel, and place roasting pan in the preheated oven. Fill roasting pan with boiling water to reach halfway up the sides of the baking dish. Bake until the flan is set and a knife inserted in the center comes out clean, about 1 hour.
Transfer to a wire rack; cool. Refrigerate flan until ready to serve.
To serve, run a knife around the edges of the pan and invert flan onto a serving plate. Spoon any remaining caramel over the flan.