All domain name registrars require you provide a real full name, address, phone number and email address because ICANN (Internet Corporation for Assigned Names and Numbers) requires it.
If you do not provide your real contact information and a dispute or complaint about your domain name arises, ICANN will try to contact you by the means provided with your domain name registration. If ICANN can not contact you, they may proceed with a judgement and you will not get a say in the proceedings. More importantly, ICANN has the power to take a domain from you and give it to somebody else.
A PO Box will suffice in place of a mailing address.
As well, ICANN requests that registrars reverify the domain registration information periodically, so you may lose the domain name if ICANN can not reach you.