Heat a heavy skillet over low heat for 2 to 3 minutes. Toast the allspice berries until fragrant, about 1 minute, then roughly crush.
Combine the rum and crushed allspice berries in a quart-sized glass jar, seal the lid tightly and give it a good shake. Steep for 10 days.
Strain the allspice-infused rum through a fine-mesh sieve into a small saucepan and add the sugar. Heat over low, stirring occasionally, until the sugar is fully dissolved, about 15 minutes. Cool to room temperature before storing in a glass jar or bottle.
Note: If you can't find navy strength rum, use 435ml of a 151 proof rum to combine with the allspice berries. Then instead of adding the sugar directly to the rum, create a syrup using ⅔ cup of water with the cup of sugar, then combine that with the strained infused rum to get the same ABV.
