In a small saucepan toss together the blueberries, peaches, water, sugar, and corn syrup.
Heat the mixture over medium-high heat, stirring often until it comes to a simmer.
Reduce the heat to medium-low, and simmer the fruit for 3 minutes, stirring frequently. The mixture will just be starting to thicken.
Remove the pan from the heat, and allow the fruit to cool to room temperature.
Blend the fruit:
Puree the fruit mixture until it's completely smooth: you can use an immersion blender to do this right in the saucepan, or puree the sauce in a blender.
Transfer the pureed sauce to a covered place it in the fridge to chill for several hours or overnight. It needs to be completely cold.
Make the sorbet:
Assemble your ice cream maker, and have ready a large freezer-proof bowl to hold the sorbet once it's done.
Add the lime zest and lime juice to the chilled fruit mixture. Whisk the mixture well to incorporate all the sugar and lime.
Pour the mixture into the bowl of the ice cream maker, and churn the sorbet according to the manufacturer's instructions. You may have to do this in two batches, depending on the size of your ice cream maker.
Process the sorbet until it has the right frozen texture. (In my ice cream maker (a Cuisinart Fruit Scoop) the sorbet was done in about 20 minutes.)
The sorbet can be served immediately, though it will be soft. Or, transfer the sorbet to the freezer-proof bowl, cover, and freeze it until it's firm.
The sorbet can be stored in the freezer for up to one week. Give it 10-15 minutes to sit out at room temperature to make it scoopable.
Makes about 6 cups of sorbet.
Notes
Allow time in the recipe for the fruit puree to chill thoroughly.
The ice cream maker I use is a Cuisinart Fruit Scoop which has a bowl that must be frozen before using. If your ice cream maker has a freezable bowl, place it in the freezer for at least 24 hours before using it.