Does Jordanian customs stamp your passport?
 

Two questions really:

1) Does Jordanian customs stamp your passport when you enter the country with a bike (without a carnet)? Or do you get a separate piece of paper? I assume they would stamp something, right?

2) If you do get a stamp for the bike in your passport, would the Jordanians NOT stamp it out when you exit Jordan with a vehicle at the King Hussein/Allenby Bridge, like they do with people?
(Apparently, at the KHB, Jordanian immigration don't give you an exit stamp because the West Bank is not considered Israeli territory and the bridge therefor not an international border crossing.)

I'll be doing Germany-Israel-Germany and would like to return via Jordan, Syria etc, which is why I want to avoid any Jordanian stamps in my first passport when entering Israel. I would then enter and exit Israel on my second passport, reentering Jordan on the first.

Thanks a lot,
Julian