I've run and played in lots of multiple-player Song of Blades/Mutants & Death Ray Guns games. Most of our games have been "every man for himself" rather than with two sides, but I think two sides makes it even simpler.
What we do is just go round the table. Each player gets a set of activations as normal. The trick, I think, is to keep the model count per player fairly low - three to five, say. One or two players might have more, but you'll get a much faster and more involved game by keeping the number of models low. I'd allow each player to have just one personality.
If the players are divided into two sides, you'd obviously want to alternate between side A and side B in the activations.
Games with several specific objectives work really well for this sort of thing. Most of our multiplayer games have been treasure hunts of some sort, with several treasure items on the table.
You can get round the problem of having players knocked out of the game early by allowing weaker replacement forces to take the field. So, if each player starts with 300 points, you could allow a replacement side to have 200 points, then 150. We found that that works well to keep everybody involved. Of course, if you want a short and snappy game, you can just rule that once you're dead or routed, that's that.
I wouldn't bother with the reaction rules (as in ASOBH) for a multiplayer game. You can make it work, but it makes it hard to remember who's actually the active player.
An alternative to having six players taking successive turns is to divide a long table into three zones and have three simultaneous games going on side by side. If a player's troops enter another zone, they are controlled by the friendly player responsible for that zone.