The overseas English teaching gig is rife with this stuff.My sister taught English in Greece for years without proper documentation. She had to leave the country every six months or so, then return.
She'd go across the border to Bulgaria or where-ever, then turn around and come back.
She got caught once between two borders and spent many happy hours trying to convince someone to let her into one or the other country.
This deal in Korea could very well be related to some politician who is trying to look like a bit of a crusader.
It's true the teachers shouldn't be working there in the first place 'cause they don't have the proper paperwork, but they will have been assured they were ok, and no one was going to bother them. Otherwise they wouldn't have gone over.
Goes to show what can happen to illegals, and how vulnerable their positions are.