In the bowl of an electric mixer, pour in the warm water and the meringue powder. Mix it with a whisk by hand or with the whisk attachment until it is frothy and thickened, about 30 seconds.
Add the cream of tartar and mix for 30 seconds.
Pour in the powdered sugar and, using the paddle attachment, mix slowly on the lowest speed for a full 10 minutes. The icing will get thick and creamy.
Cover the bowl with a dampened tea-towel to prevent crusting and drying.
Remove some of the icing to a small bowl and add small amounts of warm water (~ 1 teaspoon at a time) to reach the desired consistency - run the icing off the back of a spoon back into the bowl, and if the drips disappear into the icing in 5-10 seconds, it's good. If you have gone too far and it's too thin, add more of the original "stock" to make thicker. Color as desired.