I used to travel for work in Near-Middle east in the 90's.
IIRC, you can't ask for a vis to Syria (jordan, I can't remember), UAE, Saoudi Arabia, Koweit etc... if there is a stamp from Israel on your passport. At this time, I had to use 2 French passport (one called a Professionnal Passport with limited validity of 2 years) was to enter in Israel only so the stamps did not show in my other passport for other countries.
Having 2 passports is also very useful when one passport is at the different embassies waiting for Visa approval.
Now, I have the Canadian and French passport which makes things even simpler.
Fred, Airhead Lover
BMW R90S 74 & R100GS 93, Ural Patrol 06