My given name is Anthony, and I have this problem. I grew up in Australia, and visit some family in Poland every other year. The Australian pronunciation of my name is particularly incompatible with Polish, so I change it from 'ænθəni' to (something like) 'ɑːntɒni'. The 'æ' sound at the start of my name is totally foreign, and if I start introducing myself in Polish, then say my name the way an Australian would, the entire sentence just sounds too weird. Obviously Polish people are familiar with Western culture, and have probably seen the name before, but it just sounds too strange when used in spoken Polish.