What is a PhD in Germany?

Nature News this week reports that

Seven US-educated scientists working at the Max Planck Society's institutes are facing criminal charges for impersonating a doctor.

This extremely embarrassing story is fortunately no longer possible. The German Kultusministerkonferenz decided on March 6 (text in German) to allow U.S. PhDs to call themselves Dr. in Germany. Until this story evolved, I didn't even know that this is an issue.