This staple of the Puerto Rican holiday season needs to become a tradition of yours. Begin at least 2 days ahead (to allow for curing).
In the carafe of a 60 oz. blender, combine the coconut milk, cream of coconut, condensed milk, evaporated milk, vanilla extract, salt, and all three rums.
Sprinkle the cinnamon, nutmeg, and allspice onto the surface of the ingredients in the blender, then blend for one minute on low speed.
Use a rubber spatula to scrape down the sides of the blender, then blend for an additional 30 seconds- 1 minute on medium speed.
Add 2-3 cinnamon sticks to each of the empty bottles.
Use a funnel to pour the blended coquito into the bottles, leaving at least 1" of headspace in each bottle.
Cap the bottles and refrigerate for at least 48 hours to age.
Twenty minutes prior to serving, remove the bottle of coquito from the refrigerator to allow the fat in the coquito to warm up.
Give the coquito a vigorous shake to mix the spices that will have settled.
The alcohol in this recipe will cure the coquito for at least a year- maybe longer.
If your coquito is still very thick after allowing it to warm, try to pour it into the blender to blend it again.
Because virgin coquito doesn't contain alcohol, it will only be good for one week after blending. Make it when you know you're going to be able to finish it within that time. Virgin coquito does not need to cure the 2 days.