I would say that it depends on if there is any noticeable propagation delay thru all the "soundboard" equipment. If that is negligible (<0.1ms?), then I think it isn't that there is no phase relationship, it would just be unknown. So the soundboard might be out of phase from the stage and when you decoded the M+S you might be swapping left and right channels.
If there is some unknown level of propagation delay, then I'm with Steve -- you'll be left with junk.
If it were to work though, the soundboard channel would essentially be a omni source (not a directional, cardioid source as MS generally specifies). You can look up what happens if you use an omni as your mid mic, but basically regardless of the M level used or S level used, you end up with an effective pair of some type of directional mics facing 180 degrees apart from one another. Depending on the levels used you could effectively have subcards, cards, or hyper/supercards, but they will always have an effective included angle of 180 degrees.
I'd still be curious to hear how this played out in practice.