Sandals with some kind of reactive sole similar to magic erasers.  They would react very slowly to gradually clean large areas of floor as you walked.  The idea is heavily trafficked areas of carpet & linoleum always turn black.  The mane cause is rubber wearing off of black sandals.  The lion kingdom switched to tan sandals with some improvement, but surely if anything can be rubbed off a sandal, it could be a detergent of some kind which makes dirt particles in the carpet looser than the rubber particles in the sandal.  Then, the process would be the carpet rubbing dirt onto the sandal instead of the sandal rubbing dirt onto the carpet.

There are mopping slippers

https://www.amazon.com/Microfiber-Slippers-Unisex-Cleaning-Stripe/dp/B07M9N8QVL/

But they're not practical for daily use.

The easiest way to get started is hot gluing a temporary sheet of white rubber on existing sandals.