EDIT: My answer is based on severely outdated information, please accept my apologies for not being more thorough. jpatokal has a much more up to date and accurate answer, please refer to that. It is now possible and legal to own a voice enabled SIM as a foreigner on a short stay.
Since April 1st, 2006, the cellphone-related law called “the Act for Identification, etc. by Mobile Voice Communications Carriers of Their Subscribers, etc. and for Prevention of Improper Use of Mobile Voice Communications Services”, or in short, “Mobile Phone Improper Use Prevention Act (携帯電話不正利用防止法）” requires anyone who intends to have a voice-call-capable cellphone service (whether it is a prepaid contract or a postpaid one) to show a proper identification which indicates that s/he has the residential (permanent) address in Japan. A hotel address is not sufficient for this purpose.
This law became necessary because Japanese authorities, and especially the law makers, consider the fraud using the cellphones became a social problem, and the police and other law enforcement agencies want the address of all the phone owners to be registered so that, if any crime is committed using a cellphone, they can find the owner and search his residence.
The official proof of residence includes
– Certificate of Residence which must be registered, and issued as a proof, at the municipal office where you reside,
– Japanese driver’s license, governmental health insurance card, or anything that will be issued with the proof of the Certificate of Residence mentioned above,
– Japanese passport, with the Japanese address (hand-)written in the address page by you,
– Proof of an alien registration for a foreign person living in Japan, again, registered and then issued as a proof of residence at the municipal office where you reside. The alien registration is issued only if you have a valid visa which permits you to stay in Japan for more than 90 days.
So, as a traveler to Japan, who does not possesses a valid Japanese passport (due to a dual citizenship) or does not have a visa to permit you to stay there for more than 90 days, you simply can not obtain any voice-call-capable cellphone service or a local cellphone SIM with a voice-call, whether it is a prepaid contract or a postpaid one, under your name, period.
Apparently the PAYG SIM doesn't apply to the law, because it has data and voice capabilities (read this answer's comments for context). So if you would like a non-data/VoIP reliant phone service, you can look into b-Mobile's PAYG SIM. This provides an important advantage in that you're able to call emergency services, which you cannot do in Japan over VoIP. Could potentially be a life saver.
I'm finding it hard to understand how/why exactly PAYG SIMs aren't subject to the law, but it is as of right now a good option.