Well people, I'm officially claiming victory. No-one guessed Remember This and I pretty much dragged it out for an entire week. I'm pretty impressed with myself, but now it's time to relay the final answer.
It was The Return of Ishtar or Resurrection of Ishtar -- I would have accepted either -- an action RPG thing that ran on the Namco System 86.
This might be the most vague game we've ever run on Remember This. I doubt I'll ever reach these giddy heights again!
Anyway, enough gloating from me. We have a new thing going on today. Get wit' it.