I am currently a PhD student in my own country. I am also applying to PhD programs in the US right now. My intent is to try and work for both degrees simultaneously because I am worried about the current political climate and want to have a back-up plan in case something goes very wrong in some way in my home country in the coming years.
I have seen cases when people go to a foreign institution, study there full-time and defend their doctoral thesis back home using the research done abroad (of course, their home institution agreed to that). May this lead to problems in the future? For example, may the US institution retract the degree it granted if it becomes aware of the student having been a student at a different institution simultaneously?