You should come to the ESL Cafe teachers' discussion group, Japan section. If people aren't already teachers there, they want to be.
Bottom line:
Sadly, your nationality and upbringing may work against you, but this is not to say that it is impossible. There are essentially 2 types of work visas you can get for teaching, plus (possibly) a working holiday visa.
The Instructor visa lets you work as an ALT in public schools.
The Specialist in Humanities/International Relations visa lets you work in conversation schools (eikaiwa) and at companies that want employees to learn English.
The WHV lets you work anywhere, but it's limited to a year and to certain conditions of age, funding, and such. Plus, you get taxed more (20 percent).
MOFA: The Working Holiday Programmes in Japan
The first two visas usually require of non-native speakers like yourself that you prove you have had 12 years of your education all in English.
To get a visa, you must first be hired. After that, you apply together for the visa. But, you need to figure out first which visa you want and whether you are eligible to get it. Employer conditions for hiring can easily be different from that of immigration's requirements for a visa.
Immigration Procedures Guidebook
Application for Certificate of Eligibility
Don't bother going through agents.
Come to the ESL Cafe and see what other Taiwanese have managed to do.
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