In a small saucepan over high heat, add cinnamon sticks and water and bring to a boil. Reduce heat to medium and continue to cook until liquid is reduced to 1 cup. Remove cinnamon sticks and allow cinnamon water to cool.
In a medium saucepan whisk egg yolks and evaporated milk until well blended. Cook over low heat, stirring constantly until mixture thickens and can coat a spoon. DO NOT BOIL! Allow mixture to cool only slightly.
When cinnamon water has cooled slightly, stir in the coconut milk until well blended. In a large punch or mixing bowl, whisk together the coconut mixture, yolk mixture, sweetened condensed milk, and rum. Chill well and serve.
Notes
For Non-Alcoholic Coquito - Use only 3 cans of sweetened condensed milk and leave out the rum.