Put stemmed black currants and water into a medium saucepan and bring to a boil over medium-high heat. Reduce heat to medium-low and simmer, stirring occasionally with a wooden spoon, until berries begin to break apart, about 10 minutes.
Strain fruit through a fine sieve into a bowl, firmly pressing on the solids with the back of the spoon. Add 4-6 tbsp. sugar to coulis while still hot, stirring until sugar dissolves.
Transfer coulis to a glass or ceramic jar and set aside to cool (it will thicken as it cools). Coulis will keep, covered, in the refrigerator for up to 2 weeks.