Find a very large canning jar or purchased herb saver. Make sure the jar is large enough that the cilantro stems won’t be squished or packed inside. Place a few inches of water inside (or up to the water line in the herb saver).
Place the cilantro stems inside the jar, with the cut side in the water. (Cut off the ends if necessary to fit them in the jar.)
Screw on the lid of the jar or add the top of the herb saver. Store refrigerated for up to 1 week (or more, if using the herb saver).