1. Combine cinnamon sticks, juice drink, and ginger slices in a large Dutch oven, and bring to a simmer. Cover and cook over low heat 30 minutes. Discard cinnamon sticks and ginger slices. Stir in rum and lime juice; serve warm. Garnish with sugar cane sticks or cinnamon sticks, if desired.
Make-ahead tip: Steep juice mixture up to a week ahead; cool to room temperature, and refrigerate. Warm over medium-low heat, and add rum and lime juice shortly before serving.