However, this was a time when I had got to have a passport to live in Johannesburg.
This sentence comes from my textbook in an article about Nelson Mandela. According to Webster's Dictionary of English Usage:
Have got to, have to, and the frequently recommended must can all be used in the present tense, but only had to can be used in the past.
could we say that the 'had got to' is not correctly used in tense?
It is correct.Its equivalent in present is "has/have got to" meaning "bound to" or "can't help doing something"
"You have got to be punctual"
"A poor man has got to dream"
In your sentence, "I" was bound to have a passport.
I would not write, "had got to" - probably because I have read and heard so often that it is incorrect that I can't bring myself to leave evidence that I have produced it. But I know that I use it in everyday conversation, and I've noticed that at least some of my colleagues do so too.
I am fairly sure I always use "I'd got" rather than "I had got".