Scoop the mango and pineapple into a blender. Add the corn syrup and lime juice and blend until smooth, about 30 seconds. Pour the mixture through a strainer set over a bowl and use the back of a spoon to press the liquid through the strainer. Discard any solids. Cover the fruit puree with plastic wrap and refrigerate until very cold, several hours or overnight.
Follow the directions according to the ice cream maker's instructions and churn/freeze the sorbet until it's the consistency of soft serve -- about 15-20 minutes. Transfer to a container and freeze for two hours or until set.
Can be served straight from the ice cream maker -- as a soft-serve. If frozen overnight, let the sorbet sit on the counter for 15-20 minutes before scooping - to soften up a bit.