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.
Any overproof rum will work here, meaning any rum over 50 percent ABV. But navy-strength rum, typically bottled at 57 percent ABV, is my favorite. Just remember: The higher the ABV of the rum, the higher the ABV of your resulting dram.