For a while, at least through out most of the 70s, 80s and 90s, names were romanized using castrated Wade-Giles, meaning Wade-Giles without any diacritics. WG isn’t very useful without diacritics.
Today people get to choose their own preferred spelling. If you don’t specify your preference, the clerk will just assign their own preference to you. For the last 10 years, most people in the greater Taipei area probably got a Pinyin romanization.
Sometimes people insist on using WG because it’s a hassle trying to explain to customs that you are traveling with family when the dad’s last name is spelt Tsai and the children’s are spelt Cai.
I don’t recall seeing anyone with a TY romanization name. I’m sure they are out there though.