Cut the green leaves from the strawberries and rinse under cold water.
Pat the strawberries dry with a paper towel.
Place the strawberries in a blender with the blackberries.
Add the water and blend on high until completely pureed.
Pour some of the mixture through a sieve into a medium bowl and start to push the liquid through with a spatula.
Discard the leftover puree and repeat the process until all of the liquid from the blender has been passed through the sieve.
Add the honey to the strawberry mixture and whisk to dissolve.
Taste and add more honey if desired.
Serve at once over ice or store covered in the refrigerator until ready to use. Can be made a day ahead.