I use cotton bonnet as well.I do with a spotting towel or cotton bonnet. Sometimes my shirt.
Iāve recently started carrying two flat mops for this very reasonI always mop and use an orange contractor dust mop on the tiles after I clean carpets. Particularly hallways.
It always surprises customers and leaves them with a positive.
I'm pretty sure I don't dirty the tiles but you'll always get a small amount if water on them.
Who does this as well?View attachment 95406