Before you begin
Which of these is you?
This is based on your location. Where you are (each of you) and where in Japan the marriage is registered (usually the city where the Japanese spouse lives). Pick the closest match — each procedure shows you the order things have to happen in, how long they take, what they cost, with useful information such as what may hold up the process as the process moves forward.
The textbook case, and the one the Certificate of Eligibility was designed for. Your spouse is settled in Japan with an address, a job and a tax record; you are abroad waiting to join them.
You hold Engineer, Student, Dependent or a similar status, you have married a Japanese national, and you want to switch to the spouse status — usually because it removes the restriction on what work you may do.
A common case: a US citizen married to a Japanese national, both living in the United States, who now want to relocate to Japan together. It looks like the simplest scenario and it is quietly the most awkward one, because the Japanese spouse's own position has to be sorted out first.
You came to Japan on a short visit, married while you were here, and would rather not leave. This is an extremely common situation and the answer is not the one most couples hope for.
You are living in Japan on a spouse visa, the residence card expires soon, and the marriage continues.
You held the wedding and completed the paperwork in the foreign spouse's country, and nothing has been filed on the Japanese side.
A child has been born to you in Japan
acquisition
You are a couple already in Japan and a child has just been born. There is a deadline on this one, and it is short.