Hi @a_2023
There is a travel document called a ‘Reiseausweis für Ausländer’ which can be issued by the Ausländerbehörde. It is sometimes referred to as a grey passport (grauer Pass).
The Ausländerbehörde only issues this in cases where it is unreasonable or just not possible for a person to obtain a passport, e.g. if their embassy does not currently issue passports and the person has no way of having a passport application in their home country processed. I’m afraid that the fact that your embassy is taking time to process your application is not a sufficient reason to be given this type of document.
Best,
Éanna