I'm a Hungarian citizen by birth, moved to the UK 3 years ago. Later I came out as transgender and changed my name here via Deed Poll to an English name, which is completely different from my Hungarian one (both first name and surname). I use this name here as my official name on all my documents and records (bank, DWP, etc.) and consider it my real name. I want to acquire British citizenship when it becomes possible (around 2018-2019 because of other reasons) with the English name I use everyday.
My passport will expire in 2016 and I want to change my Hungarian name and official gender marker on my passport before that. My problem is that in Hungary they don't allow people to change their name to foreign-sounding names and I'd have to pick a first name from a government-regulated list of Hungarian names. But the reason why I picked an English name in the first place is because I don't like any of the Hungarian first names, including the Hungarian equivalent of my English name. I also want to change my Hungarian surname, because I broke ties with my family, but my English surname doesn't have a suitable Hungarian equivalent. After some thinking I selected a new Hungarian first- and surname I'd be fine with on my Hungarian passport (I wouldn't use it for anything else, living in the UK.)
I also want to keep my Hungarian citizenship and my Hungarian passport, which only seems possible with a Hungarian name.
So my question is: Is it legal (or even possible) to have two completely different names on my two future passports?