Changing jobs to a non-Japanese company?

I already asked the immigration office by mail (waiting for reply) and probably will call them later this week but I was wondering if anyone here changed jobs (same line of work) to a company which does not have a branch/office in Japan?

In case of changing jobs within the same line of work (i.e. engineer into engineer) between companies which both have branches/offices in Japan I know you just need to hand in some simple forms at the immigration office and you're done. Does that also apply when your new company doesn't have such an office (i.e. I just write down the overseas address)? Or do I have to apply for a self sponsored visa (yes I know that's not the technical term)?

So then where will you be working?

Work from home and/or coworking spaces. I'm a software engineer so that's not a problem.

I've heard a lot about "self sponsoring" where you just fill out all the paperwork for yourself and supply proof showing you can make a living by supplying contracts (~250k yen per month) etc.

What I don't know (and will ask tomorrow at the immigration office) is whether one of those contracts has to be from a Japanese companies. How I see it, such a requirement doesn't make sense as long as I pay taxes here but hey who knows....

"Engineer" requires a contract with a company or other organisation based in Japan.

Activities to engage in services which require technology and/or knowledge pertinent to physical science, engineering or other natural scientific fields based on a contract with a public or private organization in Japan.

"Self-sponsorship" is basically where you set up your own company to sponsor your visa. See here for example: http://www.tokyodev.com/2014/04/17/creating-company-japan/

Japanese immigration authorities issue visas/resident status based on "if needed" basis, I'd say. You apply for a student visa because you need to be in Japan to attend a college/language school, you apply for a work visa because you need to be in Japan to be able to work for a company in Japan that wants your skills. You apply for a spouse visa so that you can stay together with your Japanese spouse in Japan.

And those rules were established before the work-from-home or remote working idea came into existence. So there isn't any visa type that simply allows you to do what you want to do. So...

I have heard of self-sponsoring, but this seems to require that you've worked for some years on work visa (meaning working for a Japanese company) and that you want to continue similar work but on an individual contract basis with multiple companies, so I don't know if this applies at all.

@Firas, this is super helpful, thank you! Setting my own company sounds like a lot of work... But I guess that's better than nothing.

@AK, yes that's what I've heard also! I've been working as an engineer in Japan for over 2,5 years now and I'm fed up with the companies here but I want to stay because of my GF plus the company I'm entering is considering setting up a branch here if we find more people... But the again the links Firas posted pretty much shatter all my dreams...