Pour the water into the blender. Add the frozen berries, lime juice, honey and ginger. Blend until smooth, stirring as necessary.

Set a fine mesh sieve over a bowl or large measuring cup. Working in batches, pour the berry mixture into the sieve and push through with a rubber spatula.

Pour the agua fresca into a pitcher for serving. Divide the mixture between 4 glasses. Top with extra water or sparkling water (if desired), and stir to combine. Garnish with lime wedges and serve.