Here is an authentic recipe for Puerto Rican coquito. This recipe has no eggs and is uses coconut milk. This is very popular during the holidays and sometime it is called egg-free eggnog.
Place the water, sweetened condensed milk, rum, vanilla in the blender. Add the coconut milk last just up to the 6 cup line. Hold the lid tightly and gently blend. Chill in the fridge for up to 6 hours before serving. Stir again before serving.