Good luck! I think fewer people will strike than you think. There are tons of people that know they are mediocre and overpaid, so they aren't going to rock the boat. They're the ones who've had the same title for 5+ years and are nobody's go-to person for anything other than menial stuff.
In the battery of trainings managers receive, part of it is how to handle discussions about unions. Basically they're supposed to report it to legal right away. They (corp GF) also continually monitor the "climate" and other unions in the surrounding area. Another tidbit from that training was how they dealt with people talking about unions in the past. Apparently they talked out their grievances, but I'm not sure if only those individuals benefitted or larger groups of people did. This info is like 4-5 years old.