One now wonders what the Saudi SOFA looks like.
They typically get arrested by local law enforcement but whether they get prosecuted I believe depends on the severity of the crime. Obviously nothing new there. It's not unheard of for individuals to get in trouble, and with the increase of their enlisted going through USN training, it doesn't just happen at NPA.
While certainly not all Saudis, there are individuals who will prolong their training, well before even making it to API, just so they can stay in the states. It's not uncommon for individuals to fail language tests to stick around a college campus INCONUS until finally the Saudi government sends them home.