In one bowl, whisk the coconut milk, half the honey, half the rum, and half the mint together. Place this bowl in the freezer while you prepare the other lime mixture.
In a separate bowl, whisk the lime juice, other half of the honey, other half of the rum, other half of the mint, and the mineral water together.
Either alternate spoonfuls of the lime and coconut milk mixture into popsicle molds or combine both bowls and fill molds, leaving ½ inch at the top.
Insert popsicle sticks at some point during the freezing process.
Freeze for 10-12 hours.
Serve and enjoy!