Using a blender, pureé the drained fruit and shredded coconut until smooth and set aside.
Add 2 cups (480 ml) of lychee syrup from the cans into the blender. Mix quickly with a spoon. Keep the blender in the fridge for a couple of hours to allow the flavors to meld.
Pass the mixture through a sieve into a large bowl (that will fit in your freezer). Then put the remaining solids back in the blender and blend again with 1 Tbsp of liquid. Pass the mixture through a sieve again and chill the bowl in the freezer for about a half hour (or a little more if you have time).
Right before making sorbet, add the fresh lime juice into the mixture and stir.
Start mixing with your ice cream maker according to manufacturer’s directions (roughly 25 minutes).
When it’s done, transfer it to an airtight container and keep it in the freezer for 4 hours.