Say, for example, you were a low level drug dealer and the court wanted evidence from your phone to convict your boss/supplier/whoever. They could compel you to unlock your phone, but could not use the evidence on it against you, only against your supplier, thanks to the 5th Amendment.
It's a risky option to take, as prosecutors will then ignore the immunity agreement and put you in jail anyway. I mean they tried that vs. Bill Cosby even though he had access to high priced lawyers. A regular person would have been screwed.
I think a lot of people have this idea of courts being highly technical and finding cases with precision, but in real life it's a lot more lazy and stupid people trying to avoid having to do work while still getting their job done. A great many miscarriages of justice come down to "the guy didn't want to do the work and lied about it, and nobody wanted to make the effort to verify what he said". There is a pervasive attitude of "that guy must be guilty, even if not this crime then of something, I'm not going to let him go over a mere lack of evidence. Society will be better with him behind bars."
Legally speaking, I don't think they can compel self-incriminating testimony simply by giving you immunity. They can certainly encourage it by offering immunity in exchange, but that is not the same as compelling it.
If you invoke your Fifth Amendment right not to self-incriminate you can be served with Statutory Immunity and then forced to testify, with contempt charges if you fail to testify.
There are also several different types of voluntary immunity, which is what you're describing. Those can't be compelled because they may not prevent the government from using your testimony against you in other venues, statutory immunity does not.
Hmmm . . . that is the first I have heard of it. It appears that Kastigar v. United States[1] may be the relevant case law. I think it a stretch to imply that it derives from the 5th amendment, however, as the grandparent seemed to do. It seems more a determination that the practice does not violate the 5th amendment.
Say, for example, you were a low level drug dealer and the court wanted evidence from your phone to convict your boss/supplier/whoever. They could compel you to unlock your phone, but could not use the evidence on it against you, only against your supplier, thanks to the 5th Amendment.