RockClimbingHangout.com

Well lots of people do deep-water soloing, which is basically climbing without any protection, above deep water, so if you fall you should be OK.
These climbers use normal rock climbing shoes because rock climbing shoes are the best for rock climbing, surprisingly. They go straight from the water to the rock.
Rubber is hydrophobic so the grippy sections don’t stay wet for long, and any dust or whatever on the rock usually dries them pretty quickly anyway.
so, just go for rock climbing shoes! and you should be able to find these at most climbing walls, or outdoors supplies stores.

There is no perfect shoe for both rock climbing and water use.

All climbing shoes have a leather upper (I don’t know why) and this is not ideal for the water as the leather will dry very slowly and will dry hard. As for the “tread” - climbing shoes have a very unique hard “sticky” rubber sole that is required for climbing and only found on rock shoes.

Most water shoes have a perforated, net, or neoprene upper to allow for draining and quick drying - I’ve never seen a climbing shoe with that kind of upper.

If you are doing very easy climbing, use water shoes with the stiffest sole you can find. If you are doing harder climbing, use climbing shoes.

Good Luck

you would want to look for a shoe that would be recommended for deep water soloing which is where you climb without any ropes above deep water. some shoes might be specified for this kind of climbing.