In a medium bowl, mix the ginger, cinnamon, cumin, anise, and cayenne pepper. Rub the chicken with the spice mixture, and place in the bowl. Cover, and refrigerate 20 to 30 minutes.
Combine 2 cups of water and basmati rice in a saucepan, and bring to a boil. Reduce heat, cover and simmer for 20 minutes, or until tender.
In a small saucepan, mix the mango, orange juice, lime juice, and honey. Bring to a boil, reduce heat, and simmer for 5 minutes, stirring occasionally. In a small cup, mix cornstarch with 1 1/2 tablespoons of water until cornstarch is dissolved. Stir into mango mixture, and simmer one minute, or until sauce has thickened slightly. Stir in dark rum.
Preheat an outdoor grill for medium heat. When grill is hot, brush the grate with oil.
Grill chicken 6 to 8 minutes per side, until no longer pink and juices run clear. Serve over the cooked rice, and top with the mango sauce.