For each serving, blend ½ cup orange juice, ½ cup pineapple juice and the 1 peeled, chunked banana in a food processor until smooth.
Pour the mixture into a freezer-proof container, seal, and place in the freezer until frozen.
Repeat the above process for each additional serving.
When ready to serve, remove as many containers as needed from the freezer and break the frozen mixture into pieces. Spoon the pieces into a food processor or a blender.
For each serving, add 2 to 3 tablespoons of orange juice to the blender and blend until the mixture is slushy.
Stir the slushy and pour it into glasses.
Garnish each glass with an orange slice and a maraschino cherry.