Having lived abroad for some time I can sympathize with the problems mindflayer and his girlfriend have. Health insurance is often a pain as a foreigner and immigrant, Germany doesn't seem to be an exception.

I looked a bit around and it seems that the OP didn't research all options (language barrier and bureaucracy is often a problem here).

@Mindflayer: I suggest you find an expert that can guide you through this jungle.

But what I found in my quick research is that there is a european health insurance card (apparently one can get it from one's EU health insurance in one's home country, it's called EHIC). It allows a German doctor to book services against your Italian health insurance. Unfortunately, it only covers emergencies.

For other medical services you are right, it's probably easiest if you have a job or are married. But it seems that there is another option: your girlfriend registers as unemployed/job seeking. Job seekers from EU countries are treated similarly as employees and hence are entitled to public health insurance.

Source: https://www.krankenkassen.de/meine-krankenkasse/krankenversi... (German)