In Japan, I used the katakana-ized version of my English name. This was more necessity than choice, as most/many places couldn't enter non-Japanese characters into their databases (including, for instance, my national insurance card).

In China, it's very common IME for foreigners to adopt a Chinese name, which often bears little or no relation to their original name. In my case, my Chinese professor in undergrad spent several days contemplating a "good" name for me, and I've used it ever since. I've gotten complements on how auspicious it is, so apparently she did a good job.