Anyone please feel free to correct any misconceptions I might have had below, if you have more examples, let me know too:

Keep in mind that stuff like this is expected to vary a lot depending on design, temperatures, solar irradiance, humidity, shapes, water flow, wind speed, wind direction, drag, etc

Based on "Design integration of novel porous ceramic evaporative cooling systems", their design had a water consumption ranging from 0.4 to 0.8 l/h/m² (liter per hour per square meter of surface area) (whereas predicted data from simulation was around 0.4 to 1.1 l/h)

For comparison, misting systems, which can be a popular technique to improve outdoor comfort in public spaces, might use around 0.8 l/h per misting nozzle

On another study, "Experimental study of cooling effects of a passive evaporative cooling wall constructed of porous ceramics with high water soaking-up ability" DOI, shows an evaporation rate of 20.7 kg/day for an area of 1.1m², converting to l/h/m², it is around 0.784 l/h, where it had max-min air temperatures at 36.4 and 26.2 °C, relative humidity (RH) at 72% and mean wind speed at 0.6 m/s

Another experimental study using porous ceramic water containers measured cooling capacity approaching 225 W/m² of exposed wet ceramic surface under hot and dry conditions (30–35°C air, 35–50% RH).

Computer modelling and experimental investigation of building integrated sub-wet bulb temperature evaporative cooling system DOI

A separate study, from the same author at different conditions, with air temperatures up to 45°C and RH up to 50%, found up to 280 W/m²:

Investigation of an Evaporative Cooler for Buildings in Hot and Dry Climates DOI

Considering latent heat of vaporization of water is about 675 Wh/kg at 30°C, it would give us an evaporation rate of around 0.333 and 0.415 l/h for the previous 2 studies respectively

If we consider that, on average, a healthy adult goes between 4-10 times per day to the toilet and that each time you flush, you can use between 4 to 22 liters of water depending on the efficiency of your toilet... yeah, I think we can agree we might have a bigger problem with water usage somewhere else

Summary of water usage below assuming the system is up 24 hours per day, which was not the case in all above examples

So if you haven't already, and if you can, change your toilet with more efficient option. Bonus points if you get a bidet

TLDR: You are more likely to waste more water going to the toilet than having a passive evaporative cooling wall