The court reporter keeps a contemporaneous, verbatim, shorthand record of the proceedings (and often has an audio tape to refer to as well), and the official copies of exhibits admitted into evidence are also maintained by the court.
But, the final version of the official transcript takes at least a couple of days per day of testimony the court reporter heard to finalize (and, of course, a court reporter doesn't necessarily work transcribing day after day on one case, the court reporter may have other jobs to take down testimony at the same time that an official transcript based upon testimony that the court reporter previous took down in shorthand is requested, it can often take weeks or months).
Sometimes drafts of official transcripts are provided to the testifying persons or counsel for review to catch errors in transcription, before being finalized. I don't know what Minnesota practice is on that fine detail in criminal trial transcripts.
The official transcript is used for purposes of appeal (and usually only actually prepared in final form only if there is an appeal) and sometimes for post-trial motions, but is not prepared in time to be available to the jury, and is quite expensive (several hundred U.S. dollars to single digit thousands of U.S. dollars per typical day of testimony).
Given the advances of technology, it should be easy these days to
produce a printout pretty quickly. Is there a case to made to start
doing this even though it might have been impractical in the past?
It isn't unheard of (especially in big dollar commercial cases) to have a private court reporter making a working copy of a transcript that is available in real time or by early evening after each day of testimony, but these working copies are not done to the same standards of accuracy as official trial transcripts for purposes of appeal and are done at a high price premium by elite court reporters who can transcribe faster than the average court reporter can.