Add the chopped almonds to a frying pan set to a medium heat, and allow them to gently toast for 5 minutes until fragrant and golden. Remove from the heat and allow the almonds to cool down.
Place the almonds into a large jar or bottle, and add the overproof rum. Allow the rum to sit and infuse for 24 hours.
Zest 8 limes and then add the zest to the container with the almonds and the overproof rum, along with the cloves, the allspice berries, and the fresh ginger. Mix everything well and then allow the mixture to sit for another 24 hours.
Place a muslin cloth into a funnel, and strain the almonds, lime zest, cloves, allspice, and fresh ginger out of the Falernum mixture. Discard the strained materials.
In a small saucepan, heat up the cane sugar along with the juice of 4 of the limes and 1 cup of water. Keep stirring until all of the sugar has dissolved. Allow the lime syrup to cool completely then combine it with the overproof rum mixture, stirring well. Allow the mixture to sit for another day or so before using in your favorite cocktails.