My wife has been working in Swaziland for the last year and she's getting ready to come back to the US. She just found that her passport was mis-stamped on her last South Africa–Swaziland crossing: the border agent mistakenly gave her only 7 days (apparently they missed her work permit or something, I wasn't there). She didn't notice that it was off until just now; she's now a day over the 7 days and one day from her flight home.
At the moment she's labeled (apparently) as "undesirable" and can't re-enter South Africa for a year, at least until she appeals this. She has 10 days to file paperwork for that. She will do that, of course, because she'll want to travel there again. But the concern at the moment is being able to get home. Will they let her connect through Swaziland on her trip back to the US? There is only one airport in the country and it flies only to Joberg, and the country is bordered on three sides by South Africa (and on the other by rural Mozambique... which I guess is a last-ditch possibility.)
Edit: I now have the "DECLARATION OF FOREIGNER AS AN UNDESIRABLE PERSON" form. It matches what I said above: all but "(h) You have overstayed by 1 days, at a time." and "(i) Your undesirability status will be lifted and will expire on 24/06/2018 (DD/MM/YYYY), at which date you will only be allowed to re-enter the Republic." are struck out.
Fortunately the appeals can apparently be emailed. But I'm sure they can't process them in a day, can they? So the fundamental issue remains.