I agree with Patsy. Glyphosate will get rid of it, provided you regard it as a three year project, and check for any new growth weekly during the growing season. It's rather like the alternative treatment, which is to pull the stuff out as soon as it appears, weekly, but more effective.
Mare's Tail is a bit like bindweed in that it has an extensive root system that stores energy. This energy is used to send up new shoots. You need to kill the root system by exhausting it, and that means denying it any opportunity for photosynthesis. If it receives sunlight it creates new energy (glucose - which turns to starch) and whisks it down into the root system and out of reach. So you can't let any of it escape your attention for any amount of time. the advantage of glyphosate over pulling by hand is that the plant draws the glyphosate down into the root system, maybe for 12" - 18" before it is used up, so new growth has to be pushed up a longer distance before it gets to the top (and to daylight!).
If you don't keep 100% on top of it, it will defeat the glyphosate, which is why so many gardeners say that glyphosate doesn't work on Mare's Tail, Bindweed, Bear's Breeches and other weeds with deep energy storing root systems. But if you are relentless in your pursuit of it, scouring the ground for any new growth and zapping it right away, you should be able to get rid of it. Unless you are like one poor soul whose Mare's Tail is coming across from his next door neighbour's infested garden!