Originally posted by: AndrewR
It's inaccurate to state that they never have authority off base. If the government declares a site a national security site (cannot recall the exact terminology), military police/security can secure that site with full law enforcement authority. So, for instance, if an F-117 crashes, that crash site is declared a national security site, and the nearest Security Forces/MPs/PAMO will be there guarding the wreckage until it's salvaged.
So, if you see a bunch of guys off base guarding something with automatics weapons while wearing BDUs, don't assume that they can't touch you and try to waltz pass them. It might turn out very badly for you.
Further, OSI (Office of Special Investigations) for Air Force (CID for Army and NCIS for Navy/USMC) have the authority to conduct surveillance of military personnel off base, though I suspect that they coordinate with local LE to let them know what's going on. I would also suspect that they want to apprehend someone off base, they must either seek approval or have local LE there with them -- likely the latter in case civilians are involved.