Squeeze blood oranges and tangerines until you have 2 cups of juice.
In a heavy bottomed sauce pan, stir citrus juice, sugar, cardamom, and rum. Over high heat, dissolve sugar, stirring constantly. When sugar has dissolved, remove from heat. After letting the liquid sit for 15 minutes, strain out cardamom pods and seeds. Cool completely, even overnight if you don't yet have time to slowly freeze the granita.
To freeze the granita, pour the liquid into a shallow baking pan. Freeze for 30 minutes. Remove the pan from the freezer and stir the granita to break up any frozen bits. Make sure to scrape the edges of the pan—this is where the freezing begins.
Continue to freeze for two to three hours, scrape and stir at 20 to 30 minute intervals.
Once the granita is ready to eat, remove from the freezer and serve immediately. It melts quickly.