I use a small Glad sandwich container at the bottom of the sink during 100% of the shave.
The reason to use a smaller container is to reduce foot print of soap scum and not use as much water, also to protect my razors from hitting the sides of sink when trying to rinse of suds.
Between passes you can scoop water out of the small container or just use the tap and let it overflow if you like cold water rinses it still leaves a small foot print.(Shave from start to finish after cleaning 15-25 minutes
What usually happens is fellows fill the whole sink up and do their shave pull the plug and let the scummy soap & whiskers and facial oils coat the whole basin up. With the smaller target basin it just overflows and reduces the foot print of the area that might get dirty. Also after the shave I place my lather bowl inside the plastic container to reduce clutter. I have been doing this routine for about 3 years and I do not how it can be perfected any better.
To wipe up I use All purpose towels that dry quickly and can be used many times. I will take soap out of the brush and place it in the plastic bowl and rinse my brush in there also to create soap water to clean the reusable towel and then place out to dry. Seems to work, I will not use a face wash cloth to wipe up the counter top after the shave and clean the mirror up a little. Then every 3 weeks the counter and sink get a very good cleaning. The plastic bowl is like a magnet for soap scum over time.
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Have some great shaves!
being doing this for a long time - I found a nice melamine rice bowl, fancy.
-jim