Bring liquid to a boil, and cook over medium heat until syrupy and reduced by half, for 15 to 20 minutes, watching closely toward end to prevent scorching.
Add reserved rhubarb and crystallized ginger, and cook, stirring, until rhubarb is soft and mixture has thickened, for about 5 minutes.
Pour into hot, sterilized half-pint jars, leaving ¼-inch headspace.
Tighten lids, and process in a boiling-water bath for 10 minutes.