Crush the allspice berries in a mortar and pestle or grind them in a spice grinder. You want coarse, large pieces and not a fine grind.
Place the crushed allspice in a sealable glass jar and pour the rum on top. Seal the jar and shake well. Let this mixture steep for 4 days, shaking daily. On day 5, break up the cinnamon stick and add it to the mixture.
After 12 days total steeping, strain out the solids through a fine-mesh strainer. Then strain again through a coffee filter into your final bottle or jar.
Heat water and sugar on medium until boiling, stirring to dissolve, about 5 minutes. Let the syrup cool, then add it to the strained allspice infusion. Shake and then let rest for a minimum of two days before using.