Place the first 6 ingredients in a large saucepan over medium-high heat; bring to a boil. Reduce heat; simmer for 15 minutes or until the plums begin to fall apart, stirring occasionally. Place the pan in a large ice-filled bowl; cool completely; stirring occasionally. Discard allspice.
Place the plum mixture, lime juice, and basil in a blender; process until well blended. Press the plum mixture through a fine sieve over a bowl, and discard the solids.
Pour the mixture into an 8-inch square glass or ceramic baking dish. Cover and freeze until partially frozen (about 2 hours). Scrape with a fork, crushing any lumps. Freeze for 3 hours, scraping with a fork every hour, or until completely frozen.