Three cables (nine separate wires) come into the outlet box. One cable supplies power,  another cable carries that power on to other outlets, and the third cable goes to the wall-mounted switch. This is a middle of the run installation. All of the romex lines are 2 wire. There is no red wire.

I have the white wire within the cable that goes to the switch  “made hot” and marked black at the outlet box. I have pigtailed all of the black wires together omitting the white wire that is marked black. The hot pigtail wire is attached to the lower brass screw of the outlet. The white wire marked black tape goes is on the upper brass screw of the outlet.

The tab between the two brass terminals of the outlet is broken.

The white wire of the romex going to the outlets down the line is on the lower silver screw and the white from the romex hot line is on the upper silver screw.

After the outlet and switch were wired and the circuit turned back on, I had power on both wires at the switch and at the plug with the black pigtail, but not to the plug with the white wire. The black and white at the switch are hot, but neither dies when the switch is turned on or off.

I changed out the receptacle to a different one, with the tab broken in both situations. And I changed out the switch to be sure also with the same result.

The same situation occurred if I reversed the black pigtail and white wire going to the switch on the outlet, whichever plug has the white wire that goes to the switch doesn't have power.

Is it possible that the white wire at the switch is hot because it's is getting power from the black wire at the switch that is attached to the feed in the outlet box, but when it leaves the switch and returns back to the plug it's damaged inside the wall causing it to be dead on arrival at the outlet?