This sounds weird, but you might try an overall scrub with a shampoo designed for oily hair. Follow it with a scrub of full strength vinegar and then hose it down.
At least it won't harm the finish of the floor, and it shouldn't take long to do. (don't buy a shampoo with added hair conditioners- go cheap.)
Another thought would be toothpaste and a good, flat scrub brush.
Once it looks the way you want it, you can use regular shower floor products.
I do our tile like this maybe once a month, as my darling hubby enjoys using artisan bar soap- which over time, leaves residue, especially on the floor. My own hair conditioner does, too.
I have even used harsh shampoo to pre-wash away hairspray speckles on the tile floor in our bathrooms. Most floor cleaning products leave those speckles behind.
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