Yeah.. getting names right can be quite tricky..
Here in IT we did the following (we were married in RU):
Passport with maiden name went 'lost'. Applied in RU and received new passport.
Of course they transliterated her new family name 'wrong' but since they used the exact version as shown on my RU visa which backed up the marriage certificate there wasn't much we could do.. (gifts didn't seem to work).
In the meanwhile I had already applied for immigration papers in IT using her maiden name since it would take a long while to receive her new passport. We 'found' lost passport and used it for visa application at embassy while waiting for new passport to be issued.
Cleared customs in 30 seconds with the passport with visa. We didn't mention the other one with name that was different from both of our names.
Arrived here with maiden name as per passport / marriage certificate.
Completed Residency Permit using passport with maiden name.
At the time this was done, my wife was still registered in RU because of her property there. The RU embassy in Rome had no authority to issue a corrected passport since such changes or reissue had to be done in RU at the responsible office where she was registered.
Later had name changed on residency permit to wrong name.. her passport had gotten 'lost' again and her new passport that was sent had a transliterated name that was neither mine nor hers.. This raised some eyebrows, but since we were 'known' at this office and the picture was exactly the same one as on the residency permit they couldn't find any grounds to refuse.
Returned to RU and with our newborn, birth certificate issued by the RU embassy that they did have authority to issue showing our sons family name not transliterated because it was based on an IT birth certificate. We stated that somehow the new but wrong passport was 'lost' but in any case the name had to be changed anyway to the same as our sons family name otherwise she would not be able to get him out of the country as she could not really prove he was her son.. In addition who at the passport office really wanted to accuse the RU embassy in Rome of not doing their job correctly... they hemmed and hawed but the logic was plausible and no one wanted to call the embassy so finally reissued her a new passport with our correct family name.. they still mumbled that use of the 'correct' name was really 'incorrect.. whatever.. it was done.
Returned to IT using wrong, 2nd 'lost' passport that had been luckily again found otherwise IT customs would have caused problems as the name on her residency permit was different than 2nd 'new' passport with correct name.
Finally had the residency permit in IT changed again, for the last time to our correct family name which also matched the name in her passport.
Really was'nt too big of a deal ..
Luckily here each city has it's own immigration office where you can work out such things face to face. I think if we were in US or elsewhere else we'd have been basically SOL.