Place water, sugar, vanilla, salt, allspice, and plums in a large saucepan and bring to a boil over medium-high heat. Reduce heat and simmer for 15-20 minutes or until the plums start to fall apart. Stir occasionally. Remove the pan from heat and place the pan in a larger bowl filled with ice water. Let the contents cool to room temp. Discard allspice berries.
Place the plum mixture in a food processor along with the lime juice and basil and process until well blended and there are no large pieces remaining (it should be about smoothie consistency). Pour the mixture into an 8x8 glass baking dish (or a rectangular dish works well too). Cover and freeze for about 2 hours. Remove from the freezer and scrape it with a fork and break up any large chunks. Place back in the freezer and freeze for another 3 hours, scraping with a fork each hour.* Serve in small bowls, garnished with a basil leaf.
*If you are strapped on time, like I was when I made this, you can also scrape the mixture after two hours and then let it freeze for up to 8 hours before scraping again. Just note that it will be harder to scrape and there will be bigger chunks to break up if you don't scrape it every hour.