How to remove stubborn iron stains from ceramic shower tiles?

Our water system is from a drilled deep well which was hit by lightening several years ago causing significant damages when the surge then traveled through to our home. Ever since then we continue to experience an iron-like deposit on our shower stalls and faucets as well as porcelain sinks, washing machine etc. Most surface stains can be removed with VIM or other light abrasive cleaners or even CLR however we are unable to remove the stains that have developed on our shower stalls. Of course the tiles are white or cream so they show the stains even more so. We do not have an iron filter on our water treatment system as the water tests at low to no iron? yet we continue to develop these stains. I’m able to remove the surface buildup but there are residual stains left behind that I can’t seem to remove. I’ve tried VIM, bleach, CLR, KABOOM and numerous other acidic type cleaners.
Anyone know what this material actually is and how to remove it?? Much appreciated!!!

For one thing, I’d have my water re-tested. If it looks like iron, and stains like iron, it’s probably red-water iron in the water. There is a chemical called “Rust-Aid” which is sold at Home Depot, it’s used for power washing driveways. I’m sure that if you apply some of this full-strength to the stains, they’ll come off. Muriatic acid can be VERY dangerous to work with, the fumes are toxic and you can get a chemical burn if you get it on you – be extremely careful if you try using this. It’s really nothing more than diluted hydrochloric acid and is used for concrete cleaning, etching, etc.