Thanks; I had scrap wood to test it on but that wouldn't help with knowing how it would react to the mystery finish. However, your comment gave me a thought, that I can sign the piece on an outside edge and try finishing over that. If it fails, I can paint the whole outside edge and cover it; if it works, I am all set! Any idea on whether oil based or water based poly would work better?

Try Minwax Polycrylic. It's a waterbased product that is safe for painted surfaces, and it won't yellow over time. Be advised that in liquid form, it's a milky white, but it dries crystal clear. It comes in both spray cans and tin cans, I find that it's more easily applied from the can than the spray.

Whatever you use, test to see if it causes the colors to run on your paper appliques.

There's a good chance the backgammon board has a lacquer finish on it. It seems most times many differing finish types are compatible with each other. Until they are not. I'd be inclined to just spray a quick krylon clear lacquer coat or two on it.

I echo the comment above not to believe anything you read on the internet.

I ended up doing multiple coats of wipe-on waterbased poly on the board and then rubbing it down to satin finish with 0000 steel wool and furniture wax. It turned out pretty well, though I wish I had had the time to do more coats of poly on the outside; the finish is not as nice there. It did take a long time to build up the wipe on poly to anything thick enough. The spray on polycrylic or krylon would have been much faster!

Just Beautiful!