The main problem with it, that it is simply impossible to apply anywhere without giving my in-real-life name. And, looking for that, it is very easy to find me anywhere, and also my current employer.
Of course, I can say that it is a private thing, and ask the company to not try to find out this information, or not use this information anywhere, but then they will think:
- There is no guarantee that they will follow my this wish. After they found out my actual employer, they can simply ignore my this wish and ask them about me behind my back. This will also serve as a warning. A question like "How well this M.S. worked for you?" serves also a warning: "Hey, this M.S. wants to leave you!"
- They can think, that maybe I want to hide something about my relation with my current employer.
- They can also think, that I will search my next job after them just so behind their back, as I do with my current employer.
As an additional info: I am in Germany, where I am a foreigner. Here the companies doesn't do very often something behind others back, but from the other side, there is an informational network of bosses informing eachother from their employees. They also consider foreigners as not enough trustworthy (which is, in a mean sense, is not groundless).
How to solve this problem?
Extension: Thank you everybody the answers and also the comments - they were informative to me, even the not so friendly ones. I already know, what I wanted to know. So this question can go, if you want this. Btw, I think it would be useful for the googlers of the future and closing/deleting it would be an irrecoverable information loss, but it is your decision and not mine.