I asked the Bundespolizei on Twitter. This is their reply.
In English that reads
Your documents are valid in general. You only need the originals to cross borders. It would be useful to have a copy, and to tell them about the AZR.
AZR stands for Ausländerzentralregister, which is the federal German database of everyone who has a permanent residence or some other form of long-stay visa or asylum.
So this seems not to be a problem.
The regulation that you quoted only requires that there exists a valid passport (or substitute) for the foreigner. It is not required that the foreigner has the passport with them at all times. (Incidentally, there is a very similar rule requiring Germans to have a valid ID card (or passport), in § 1 PAuswG.) You need to be able to present it on request, but not immediately; it is OK if you store it at home, or even at an embassy, and show it to the authorities a few days later. You can’t be fined for that.
However, if the police get interested in you for some reason and you neither have your passport with you nor can corroborate your identity in some other way, there is a certain risk that the check might take quite a bit longer, or that you might even be taken to the police station. The same applies to Germans who don’t have their ID card.
Credit:stackoverflow.com‘
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