Mix guava, passion fruit, and lychee juices (we like Ceres brand) in a large pitcher. In a saucepan, combine 1 cup of the fruit-juice mixture with the basil. Bring to a boil, then reduce heat and simmer for 5 minutes. Remove from heat, let cool, and strain, discarding basil.
Pour infused juice back into pitcher, and stir in lime and tangerine juices. (Optional: Add 1 cup rum.) Ladle over ice and garnish with basil and lime.