Deep, dark purple mulberries make a beautiful, garnet-colored syrup that busts with a berry-like sweetness. The key is to use ripe berries. Cook them in water and sugar until they burst. This will release their flavor and color into the syrup.
Chop the berries coarsely either by hand or by pulsing them in a food processor until coarsely chopped.
Add the berries, sugar, and water to a small saucepan. Warm over high heat until the water begins to boil. Turn down the heat to medium-high, and continue to simmer until the berries lose their form, about 10 minutes. You may need to mash the berries with a potato masher to speed up the process.
Once the berries have lost their shape and the syrup is thick and brightly colored, strain it. Use a fine-mesh sieve to do this. Pour the syrup into a pitcher. Stir in the vinegar, and then transfer to a flip-top bottle. Store in the fridge for up to 1 month.