I'd thought it was primarily retro viruses that could incorporate into a cell's DNA (e.g. HIV) and that there were some other non-retro ones that somehow gum up the DNA repair mechanism and leave some protein expression 'bits' behind. I've read that HPV works this way, with the left-behind bits encoding proteins that promote cellular growth, sometimes resulting in cancer.

SARS-CoV-2 is not a retro virus, so I would expect incorporation in a cell's DNA not to mean that the cell is producing those virus particles evermore, but it could be working like HPV, leaving some protein encoding instructions behind.

If cells are producing SARS-CoV-2 nucleocapsid protein from instructions wrongly incorporated - and those cells are, say, in the sinuses or the throat, then those patients are going to get positive PCR (polymerase chain reaction) results even when not actively infected.

I'm just a software engineer, with the reading comprehension skills needed for that job, so this is just what I seem to be reading.

If I've gotten something wrong, I'd much appreciate being corrected.