Preheat oven to 325 degrees F (165 degrees C). Grease a fluted tube pan (such as Bundt(R)).
Mix bananas, eggs, vegetable oil, 1/2 cup water, 1/4 cup banana liqueur, and 1/4 cup rum together in a bowl until smooth. Beat in cake mix and pudding mix, scraping sides down as needed, until batter is smooth, about 2 minutes. Pour batter into the prepared pan.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 50 minutes.
Bring sugar, butter, and 1/4 cup water to a boil in a saucepan 15 minutes before the cake is done. Stir in 1/4 cup banana liqueur and 1/4 cup rum until glaze is well mixed. Pour glaze into a glass measuring cup.
Remove cake from oven and pour glaze around the sides of the cake while it is still in the pan; let soak for 30 minutes. Invert cake onto a platter.