Other than Permanent Resident status holders, all foreigners can stay in Japan for limited period of time. They have to submit extension or status change by expiration date of their term of stay.
In the event that they stay in Japan even after the expiration date, it’s illegal under the Japanese immigration law. It is recommendable to go to immigration before they find you out.
Departure Order is a procedure for overstaying foreigner. If they go to immigration voluntarily before immigration arrested them, this procedure applies to them. But if they have overstayed in the past, it doesn’t apply.
The main benefit is that immigration doesn’t detain them. They can buy air ticket, and just go out of Japan. And after 1 year of Departure Order, they can come back to Japan, if they meet some requirement of one of visa status.
If immigration finds overstaying foreigner, or it is second time of illegal stay, the procedure of Deportation applies to them. In this case, the foreigner is detained in immigration for certain period of time until they finally go out of Japan.
They can’t come back to Japan for 5 years after the deportation.
Special Permission to stay
The government can give special permission to overstaying or detained foreigner to stay in Japan even during the process of Deportation.
Ministry of Justice gives this permission based on comprehensive situation of each case. For example, the government considers if the overstaying foreigner has Japanese family or not, personal behavior, necessity of humanitarian intervension, social impact, to make this decision.
Generally speaking, it is not easy to get this permission.
The government makes guideline for the permission public.
Provisional release (Karihomen)
Sometimes immigration grants conditional release to detained foreigners during the procedure of Deportation. This is for taking care of their physical and mental health, and preparation for departure from Japan.
Detained foreigners can apply for it also. But immigration is free to make any decision. They can reject application without reason. There is more chance of success if they have Japanese spouse.
Even if the release is granted, the subject foreigners can’t work freely in Japan. And there is restriction about area where they can go.