There are lots of home remedies that work to some extent. If you want the "pro" product, check with your auto parts store for some "Rain-Away" or similar product, intended for use on windshields. There are interior and exterior grades, but for shower room use it doesn't make much difference which you use. You need to rub it onto a clean surface and buff it pretty well to get a good coating that will last for a fairly long time.

Ventilation is best. If you're getting serious fog on the mirrors, you're also getting soggy wallpaper - or moldy grout.

A sort of trick, for the really determined, is to mount light fixtures around or along the edges, and use those really hot incandescent "makeup" light bulbs. With the glass sufficiently heated by the light bulbs, fogging will be reduced some (no warranty on your installation, but I've had one setup that worked pretty well).

You can actually get "heated mirrors," with a thin layer of metal coating on the glass that can be wired to heat the glass enough to prevent fogging even under rather extreme conditions, in a way similar to the anti-ice (which is an effective anti-fog as well) used on airplane windshields. The ones I've seen for home installation tend to have a distinct "color cast" to the reflections, but if you don't mind looking like a beach bum with a heavy tan it's not too objectionable.

They've had some problems with these in public baths, because the color cast makes people think they've got a half-silvered (one-way) mirror and someone might be watching from the other side of the wall. Somehow or other, denying it makes it even more questionable, for those of suspicious inclination.

Of course the ultimate solution is to move your ablutions to the back porch in a tin tub. Granny ain't gonna carry enough hot water out to you to make steam on anything.

John
John