1) Halve and stone the plums. Whisk the sugars and water together in a small saucepan. Add the plums and simmer over medium heat, until tender but not mushy, about 10 minutes. Cool the plums in the liquid.
2) Set aside one half plum for garnish.
3) Put the diced limes and remaining plums in a large open jug. Crush thoroughly with a potato masher.
4) Add the cachaca and plum cooking liquid and fill the jug with ice cubes. Pour into chilled short glasses, allowing some ice and fruit to fall into each glass. Garnish each drink with a slice of poached plum.