This probably goes more in the Physician-Scientist sub-forum.
But yes, I'm currently doing a PhD away from my med school. I was originally accepted as a track 1 MD/PhD in the NIH OxCam program concurrently with my MSTP acceptance. So I'm doing my PhD 1/2 at the NIH and 1/2 in Cambridge with my degree awarded by Cambridge all while on a leave of absence from my med school.
Otherwise, I've heard of people whose PI left early enough in their PhD that they did most of their work at the new institution and ended up with a degree from there. They then come back for med school. My school also had a pilot arrangement where they were going to send an engineering student to the flagship State U for PhD but I think that fell through in the end.
Aside from that, there are a few well known examples in the MD/PhD community. Baylor MSTP students can do PhDs at Rice or MD Anderson as can UT Houston MD/PhDs. UCLA MSTP sends 2 students a year to Caltech. UCSF MSTP has a bioengineering tie up with Berkeley. Cornell has that Tri-Institutional thing going on with Rockefeller and MSKCC. Emory MSTP and GaTech has an arrangement particularly for engineering. Pitt sends some students to CMU particularly for engineering. UCSD MSTP students can freely choose among several independent research institutes nearby (some with their own PhD programs) such as Scripps, Salk, Burnham, etc. Stony Brook can send students to Cold Spring Harbor Labs. Penn immunology has an arrangement where some NIH PIs are appointed adjunct Penn faculty and students can train at the NIH just like at Penn. Hopkins has a formal GPP arrangement with the NIH as a separate program. Besides the OxCam program, the NIH also runs a joint neuroscience program (I think) with the Karolinska Institutet that should work (I don't know if any MD/PhD has actually tried it).
There are probably many more of these arrangements, but those are off the top of my head the ones that I'm familiar with or heard about.
