Unfortunately, there's no scheme to allow foreigners to stay as a resident if they work remotely for a company that doesn't have a presence in Japan, as I do. (I'd even be willing to pay taxes in Japan!)
Of course, such a provision is pretty rare, but it's still a bummer. I wonder if immigration policy might ever consider such situations—such a worker is basically injecting money into the Japanese economy for little cost on Japan's part. But perhaps there are downsides that I haven't considered.