Combine the berries in a bowl and blend with an immersion blender until smooth (or add to a blender and blend until smooth). Pass the puree through a fine mesh strainer (pressing with a large spoon to extract as much of the puree as possible - you should have 1 cup puree) into a freezer safe 9x13-inch pan, preferably glass. Pour the cold water through the strainer to rinse any remaining puree and press again. Discard the seeds.

Stir in the allulose, stevia, salt, and lime juice until everything is dissolved and evenly mixed. Adjust sweetness by adding more stevia or lime juice as desired.

Transfer to the freezer for 30 minutes. The mixture should begin forming ice crystals around the edge at this time. Use a for to break up the crystals before placing the pan back in the freezer to continue setting. Repeat every 15-30 minutes, or until completely frozen, about 2-3 hours.

The granita can be covered and stored at this point. Best served the same day but will last up to a week, frozen. You may want to gently thaw for a few minutes if it is frozen overnight or more.