If I were to run around in a white fur coat indoors and out I would be filthy in a matter of hours not to mention the weeks between baths my dog gets. I‘ve had a couple of dogs with white fur and day to day activities don‘t ever seem to make it look dirty (there are of course the romps in mud that require an immediate bath -but that isn‘t what I‘m talking about). If anything I would expect their fur to get dirty quicker than our hair. But I couldn‘t go three days without a shampoo with out looking noticeably dirty, the dog can look clean for weeks!
Your house is obviously filthy. Also a dog cleans their fur. Our dog does look a lot brighter after his baths but he never looks dirty (unless he has been rolling in mud or has been in the lake or the river) I am guessing because he cleans himself regularly. Our house is immaculate and the rugs get vacuumed at least once a day and the wooden and stone floors are polished and scrubbed at least every other day so he does not pick up dirt from there. I do understand what you are saying though as if I ever wear white I immediately get dirty even though it never seems that way when I wear other colours. Perhaps the dirt disappears in to their fur rather than stays on the surface!
Normally, dogs with white coats have that color for a reason. Herding dogs need the coat in order to be able to hide with the sheep and not upset them. Most white coats are coarse, and are more than one layer. The outer layer sheds dirt and water and the inner layer keeps the dog warm or shields them from the heat. It really is amazing to see how clean most white coated dogs are, regardless of being in or out of the house. My collies and shelties white fur is always snow white.
First of all, I haven't a clue how they really do it--nonetheless, the possibility caught my eye and mind so I thought I take a guess. 1. The first thing that crossed my mind was a fine coating of oil--something like ducks have to water proof their feathers so they don't soak up water. But that didn't make any sense. Any type of liquid or lotion would CAUSE dirt to stick. 2. The second thing to cross my mind was a little closer to my field of interest. Somesort of 'natural' low voltage static. You remember that type of cloth, (I don't remember if it was polyester) that would cause both water and dust to repell from it's surface? This is the direction I'm going with this answer. That cloth had somesort of natural static charge causing this. It is therefore my bottom line OPINION--somekind of natural low voltage static charge is causing this. (Wouldn't it be nice if humans had skin like that!) The way to test this is by petting the pooch lightly--did you feel or hear the static sparks? Sorry I couldn't be of futher help. BYE!