Just specify which name is the surname/family name/last name and which name is the Christian/given/first name.
The same problem comes up with non-Asian names too. On a Jeopardy show, one of the categories was "Two First Names" where for the individuals named in the trivia answers/questions there could be similar confusion. Example (not from that show): writer James Joyce.
And, in the U.S., it is not unusual for people to be listed last name first, although generally with a comma after the last name.
Travel tips:
http://members.aol.com/ajaynejr/travel.htm
Now can the ticket be re-issued without the (limited availability) seat's dropping from your grasp/claim and going to the next person on the waitlist?