I do not have this issue anymore, but tracking the solution was difficult and time consuming so I’m posting it here to increase the odds that someone googling the same issue will find a solution faster than I did.
This issue manifests in two ways. First, and most noticeable, is that you get an inconsistent framerate while gaming that may clear up for a time after restarting the game or the computer, but returns in short order. The other is that audio over any output may glitch, but audio over HDMI is much worse, and gets worse in proportion to how hard the GPU is working to render frames. If your game drops the framerate while tabbed out, the problem may go away or improve only to return when the game again has focus.
I tried a lot of things, some of which were destructive and irreversible, but to save someone else that pain the solution to my problem was to go into the bios and change the PCI link speed from Auto (or PCI Gen 4) to PCI Gen 3.
This problem is likely to appear if you have an older PC case and that PC also uses risers to connect the PCI card to the motherboard indirectly when upgrading the video card and/or the motherboard that is capable of the higher link speed. The riser is passive, and so will not report anything of its capabilities to prevent the and motherboard from setting the maximum speed they are capable of.
You could likely also solve this problem by purchasing a PCI Gen 4 riser cable, if the specific configuration of your case will allow the incompatible riser to be removed without affecting its integrity, but I have not personally tested that solution.
