Entry if married but not registered in Japan

My wife moved to Sweden a few years ago and we got married here. Due to the pandemic she has not been back for a long time but is now planning to go.

Japan still has an entry ban on tourists, but allow spouses of Japanese citizens to enter if i understand things correctly.

Because of the pandemic we have not registered or marriage in Japan yet.

My question is if i will still be eligible to enter as spouse or if we need to register the marriage there first?

Registration first. Japanese embassy in your residence country accepts it. What you have to prepare and submit will be provided by the embassy.

So your wife is Japanese, if I understand your situation.
Japanese nationals need to report any change in their marital status to the Japanese authorities - be it in Japan or through the Embassy of Japan outside Japan - within a certain period anyways.
So she should do that - either (1) go to the Embassy of Japan in Sweden, or (2) possibly if she has relatives/family members in Japan who can help her, check the requirements for reporting a non-Japanese marriage, and send the documents to them in Japan so that they can report it on her behalf. She will need to ask the city hall where her “honseki” is to find what is needed.
But first she should check with the Embassy of Japan in Sweden in any case.

Like the others mentioned you’ll need to register it first. The Koseki-Tohon is one of the requirements to get the Visa (and you need to show it at the boarder too).

Thanks for all your answers. It's great to find a friendly place with knowledgeable people!

It's correct that my wife is a Japanese citizen, though a Swedish resident now.

Can her family (with the right documents) register her marriage for us in Japan, or do we need to do it through the embassy?

Do you know roughly how long the marriage registration (by her family or through embassy) will take to complete (assuming all documents are in place and translated?)

Do you know roughly how long the visa application will take once we have the correct documents for our marriage status in Japan?

Have your Japanese wife read the section 婚姻届　3.日本人と外国人が外国の方式によって婚姻した場合. (Reporting a marriage. 3 In case Japanese & non-Japanese got married according to the law of country other than Japan.)

https://www.mofa.go.jp/mofaj/toko/todoke/koseki/index.html

Here it lists the documents needed, and they say you can snail mail them too, so maybe having her relative/family reporting it in Japan on her behalf might work too. I am not sure if she needs to write a power of attorney authorizing her relative/family member to do it.

In any case, she should call the Embassy of Japan in Sweden and the city hall where her domicile is in Japan to find out the exact procedures and the exact documents she needs. (When she calls, make sure she says that it is about reporting a marriage based on Swedish law after-the-fact in Japan, as against “getting married in Japan.”)

In either case, the documents need to travel back and forth (for reporting the Swedish marriage), it will take at least a few weeks (provided the postal service is working fine).

Visa (or CoE - certificate of eligibility) application process can take anywhere between 1 - 3 months, but it can take longer due to the pandemic. Best wishes.