No Brake Pedal When Engine Is Running

I have a 1999 Mustang V6 3.8 L, automatic car. I have a brake issue that is driving me mad. This whole issue started when I replaced the front calipers on the car. After I did so the car has no pedal when the engine is started. It had pedal when the car isn't running but once started it goes to the floor. I did bleed the brakes very thoroughly. So I then replaced the brake booster and still same problem. I have now just replaced the master cylinder, did a bench bleed on it and of course bled the brakes even doing a pressure bleed and I STILL HAVE THE SAME ISSUE. I have pedal until I start the car and then its gone. I have bled these brakes probably 5 or 6 times and like I said even did a pressure bleed. I can't find any leaks or anything that jumps out at me. Am I missing something somewhere? Can anyone help and give me a suggestion as to what this could be. I am stumped and my buddies are stumped as well. The car doesn't have ABS. But is there some kind of sensor or electrical issue I'm missing. Somebody help me bc it's been 2 weeks since I installed the calipers and the car hasn't been right since.

The calipers were replaced bc the drivers side one was sticking so we went ahead replaced both front ones at the same time. They were replaced with stock units. We've bled the brakes numerous times following the procedure you mentioned. The pedal does hold under constant pressure with the engine OFF. It goes completely away and falls to the floor once the car is started. We're getting ready to replace the check valve on the booster vacuum line. It's one of the last things were left with changing. We've replaced almost every component on the brake system trying to solve this issue.