To a medium size pan add water, lime juice, ginger root, and sugar. Stir. Heat the pan on medium heat.
When the water comes to a boil, add the cranberries and stir. Cook until the cranberries begin to pop (about 5 minutes). If the liquid in the pan is boiling too rapidly (and rising), turn down the heat to low.
Turn off the heat and move the pan to an unused burner on the stove, or on a heat resistant surface or hot pad.
Add the blackberries. Gently stir together. Top with lime zest and serve warm or chilled.
When cooking the cranberries, if the liquid in the pan is boiling too rapidly (and rising), turn down the heat to low.
As the cranberry sauce cools, it should thicken a bit.