In a pot over high heat, add the blackberries, sugar, limes, basil leaves and ½ cup water.
Smash the blackberries with the back of a wooden spoon and stir to combine.
Bring to a boil then reduce to medium heat to cook for about 5 minutes.
Remove from heat and allow to cool for about 5 minutes, then pour into a blender to blend everything together again. Strain over a pitcher to remove seeds and solids. Add in the extra 1 ½ or 2 cups of cold water and stir together (add 1 ½ cups first then taste to see if you'll need the extra ½ cup).
Serve immediately with fresh blackberries and basil leaves in each glass.
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