Marriage in Japan with foreigner

Hello, I'm here in Japan with a tourist visa and got recently married (here in Tokyo) with a Korean girl who has a business visa (3 years). Before moving back to my homeland we need to stay here for another 3-4 months so I have to apply for a marriage/spouse (or anything) visa.

In order to do so, is the marriage certificate got by the kuyakusho enough or do I need also the marriage registration certificate from my and my wife's homeland?

Thank you very much.

If I am recalling correctly from the following page: http://www.japan-guide.com/e/e2221.html

I believe that a spouse visa in Japan may require a Japanese spouse or one of you need to get permanent residency...

As far as I know, you are eligible for Dependent status.

So have you gotten married in Japan according to Japanese law? If so, whatever certificate that the Japanese authorities issues is enough to prove your marriage.

As mentioned you only get a "SPOUSE" status if you are married to a Japanese person.

Instead for this case you would get a "DEPENDENT" status.