Place berries and bananas in a large bowl and puree until smooth.
Combine fruit puree, honey, and salt in a large saucepan. Heat until melted and just simmering. Whisk cornstarch into the cold water; stir into heated berry mixture.
Remove from heat, and stir in lemon juice. Cool slightly.
Refrigerate berry mixture until cold, about 2 hours.
Freeze in ice cream maker according to manufacturer's instructions. If you don't have an ice cream maker, cool the mixture then pour it into a freezer safe container and put in the freezer. Set a timer for 2 hours. At that time, take the sorbet out and blend it up again.
After you've blended it up a bit, pour it back into the container and freeze.