Echoing
@Stu Farish statement
That is pretty much true on almost all installations. Although some are way more strict than others, some to the point of No.
Usually the Base /Post / Installation Commander has a huge if not final say in the matter, their mission dictates that. Of course there are certain agreements between state and the installations (usually it's a agreement allowing the local (state) game wardens access and enforcement abilities that they don't have unless granted within a "MOU" aka memorandum of understanding). Outside the security forces native to the Commander or Branch's native investigative agency (think NCIS, CID, etc) the jurisdiction usually falls to the FBI, which really could care less about a person harvesting game. Some installations do have a organic Game warden ( I think Ft Stewart and Ft Rucker does, not sure) type setup already, some don't. So YMMV.
But let's be brutally honest if that land is listed as a military zone/base/land/ assembly area, posse comitatus is out the window, and that branch does have the authority to arrest, detain, and put you on trial under the UCMJ. Which trust me if they put you on trial under UCMJ there is 99.99999% chance your going to jail under the UCMJ, entry grants the UCMJ to be applied to civilians, actually anyone. Or they can simply hand you over to say to ICE or other various Federal Law Enforcement agency.
I know how I would implement such a plan to zone off the borders, and honestly I think that most would agree with my method / idea. It would be a joint venture with a majority if not all of the federal agencies, that does include BLM, NPS, etc as well as the LEOs/Military zoning (Military does include the USCG in this idea). And wouldn't be cheap but probably way more effective than now. Would it require more manpower, maybe, but if applied correctly that increase would be minimal (Military strength wise I wouldn't want it to be 1 more service member simply shift manpower and bases).
Yet allow hunting the Yotes, varmints and other game. Bear in mind I like the idea of hunting on BLM/ Federal land as much as the next guy.
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The model in my mind would be the Korean DMZ approx 2.5 miles width x the length of the borders for the Military assets with a logical thought process of stewardship between Army, Navy (yes the Marines fall here), Air Force, and USCG, coupled with BP/ICE stations between the branches zones with detention center being within that 2.5 mile zone.
Article 134 and or 192 of the UCMJ already deals with unlawful entry onto a base facility and directly coincides with 10 U.S. Code § 892. (this part doesn't require more immigration judges just a few JAG officers that are trying a unlawful entry not immigration law).
So depending on the whims and flow of legal court dockets, a illegal alien could be tried under UCMJ versus the civilian court process with a verdict of deportation back to the point of entry. All Civilian Immigration courts will move inside that 2.5 mile zone. Simple idea piss off the Judges and now move to the desert, and their safety is the same BP agents, no US Marshall for bailiffs only BP or military police forces, but the US Marshals could be used if need be. That simple Idea or concept would stave off a lot of land mass back to the NPS / BLM.
But it's just a pipe dream and thinking out loud, image trying to push that thru Congress and the Executive branch. It might be easier to repeal the parts of existing laws that prevent the implantation of that process via budget reconciliation.
Just waiting to be called crazy or anti- American Facist LMAO