There was a developer comment back in beta (maybe even when 1.0 official service was running) stating that certain gear won't be restricted by which classes can actually equip it because they recognize the desire for folks to wear different gear while in town and stuff. (vanity gear) This is why the Toxote Tunic does not have a ARC/BRD only restriction.
The equip rules and looting rules are two different things. As others in this thread have helpfully tried to point out, there is no STR on the body piece which means the armor is not functionally suitable for a MNK. The only way you will be able to need on this gear is to play as an ARC/BRD because the gear stats favor their classes.
There's nothing further you can really hope to accomplish in this thread. That gear is not an upgrade for your MNK, regardless whether you want it or not. You're just going to have to win on a greed roll if you ever want to get it unless you actually play an ARC/BRD.
