I run my cooking in ultra-filtered rainwater (the filtration system is fine enough to get rid of the real nasties). The town water supply is good as far as softness, but I have seen the blue hue it casts when I fill my white enameled antique bath tub, so I tend to limit its use.
The water I've been running in my cooking cooler has been going for several months now. I don't use it all that often, but it's probably doing its 7th cook now. The only sign of cloudiness I've noticed was short term from using greasy tongs to retrieve a cooking bag. It cleated on standing and no sign of it next time I fired it up to cook.
The smaller cooking box has just had a water change because it went traveling over the weekend, but I'll keep watch for cloudiness and see how long it takes.
I wondered about adding vinegar mainly to contain any egg white should a shell crack during cooking.