Asian countries love native speakers but if you have an ESL or EFL degree, they don't really care as long as your English is good. For example, you can apply to the JET program to teach in Japan even if you're not a native speaker of English. The only requirement is to have a degree.
If you make a survey, people will always prefer being taught a native speaker of the language but don't worry about that. You just have to prove them that you can speak and write as well as a native speaker. I am a native speaker of French but I'm working really hard to become perfectly bilingual.
Good luck to you!