As a German citizen with permanent residence abroad, you are not subject to the general registration requirement and, consequently, are not required to carry identification within Germany.

However, you may apply in person for an identity card at selected German diplomatic missions abroad that have been designated as identity card issuing authorities by the Federal Foreign Office, or, under certain conditions, at any local registration office in Germany.

Note:
Please check in advance with the diplomatic mission in the district where you are staying to see if it can issue identity cards, as not all diplomatic missions are also identity card authorities.