In a heavy-bottomed skillet, dry toast (heat without oil) the star anise, cinnamon stick, and cardamom about one minute, until fragrant and lightly browned. Transfer spices to a cutting board and lightly crush with back of skillet. (If you own a mortar and pestle, coarsely grind the spices in it.)

Add the spices and the ginger to a medium saucepan with the water and sugar. Cut the vanilla bean in half lengthwise, scrape out the seeds with the tip of the knife, and add the seeds and the pod to the liquid. Bring to a boil over medium heat, lower heat and simmer until liquid reduces to three cups, about 10 minutes.

Meanwhile, cut each plum in half and remove the pit.

Add the plums to the simmering liquid and simmer until the plums are softened and skins start to slip off, about 10 minutes. Use a slotted spoon to transfer plums to bowl; when cool, remove their skins. Pour the poaching liquid through single-mesh strainer and into bowl with peeled plums; discard spices.

Working in two batches, purée the plums and the poaching liquid in a blender until smooth.

Pour soup through a single-mesh strainer. Refrigerate the soup until chilled, at least two hours and up to two days.