I think what you are looking for is a duplicate passport. I can tell you how I got one as a US citizen.
I was travelling on a motorcycle in Turkey, and needed to submit my passport for a Syrian visa at the Syrian Consulate in Istanbul, and would have to wait a week or so for them to give back the passport.
Before I left my passport at the Syrian consulate, I went to the US consulate (I'm a US citizen), and asked for a duplicate passport, because I would be applying for Visas while travelling. I was planning on going to Bulgaria while I waited for my Syrian visa to be processed. The US Consulate was fast about having a duplicate passport made. Took one day. The duplicate passport looks a bit odd, but I've used it a bunch of times and never had any problems.
One thing that also might help is a letter from an employer, saying that it is required that you travel for business while at the same time you will need to submitt your passport to apply for visas.
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--Terry
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