Sorry to chime in here so late. I don't read the forums much.
I've lived in Japan for the past 7 years. You don't have a degree, but you are eligible for a spouse visa, which permits you to do any sort of work that the employer thinks you are qualified for. That last phrase is the key, even for teaching jobs.
Spouse visa, dependent visa, student visa, cultural visa, and working holiday visa all have no requirement for a bachelor's degree, yet you can get work on them, but you still have to meet the demands of the employers. Most will require a bachelor's degree simply because they feel that graduating from college (even with a degree unrelated to teaching English) is a sign of achievement. Some employers actually write a brief biography of their teachers on their advertising materials, and not having a degree at all won't attract customers.
So, the bottom line is, you can get work, but you'll have to accept the fact that many employers will turn you down. Persistence is key, and having a good-looking resume and cover letter are tantamount. Your PERSONAL looks are not so much important, despite what your wife says. Plenty of geeky looking teachers here.