Generally a person has to have a valid passport before the authorities can deport them. However, Germany has agreements with some countries whereby the home country will accept a temporary travel document issued by Germany for the person.
This means that Germany issues a document for the person and they are then deported with it. However, it is difficult to obtain clear information on this process, i.e. with which countries it is being done, how often, and how the procedure works in practice. It is therefore difficult for me to say how likely or unlikely this is to happen in your situation.