It does not matter if the atmosphere absorbs the radiation from your device or it reaches space, after a few feet it's irrelevant. The important bit is the underside of the tarp which is not going to be below ambient temperature. Making the absolute best it can do the equiviemnt of a fan on the roof which can't replace AC.
It is essentially similar to the way in which you can freeze water in a shallow dish on clear nights that are nonetheless slightly above freezing temperature - you are facing a black-body radiator that is well below ambient local temperature.
PS: You don't actually need a thing special at night to get this effect: http://www.asterism.org/tutorials/tut37%20Radiative%20Coolin...
Of course it won't pump heat backwards into the sun but they propose a water circulation system, which I imagine could store some daytime heat to release it at night. That and masonry of the walls of the building would store even more.
"We further demonstrate the effectiveness of radiative cooling for a relatively large thermal mass using water as a cold storage medium. A plastic water tank was placed underneath the radiative cooling glass-polymer hybrid metamaterial, putting water in close contact with the heat-conducting copper plate. Since the water is stationary in the experiment, its large heat capacity substantially slows down the cooling process. We therefore used a 10-µm-thick HDPE film on top of the Polystyrene foam box in this setup to reduce convective heat loss and improve thermal isolation. The water temperature continuously dropped, reaching more than 8° C below ambient after two hours of exposure."
PS: An easy demo is a car roof with the windows open at night in a field.
You are right it won't replace the AC, but the technology could lead to a reduction in overall AC use.
Don't get me wrong it's a neat effect, just not particularly useful.
"A 72-hour continuous measurement of the ambient temperature and the surface temperature of an 8-in-diameter hybrid metamaterial under direct thermal testing. A feedback-controlled electric heater keeps the difference between ambient and metamaterial surface temperatures less than 0.2°C over the consecutive three days. The heating power generated by the electric heater offsets the radiative cooling power from the hybrid metamaterial. When the metamaterial has the same temperature as the ambient air, the electric heating power precisely measures the radiative cooling power of the metamaterial. The continuous measurement of radiative cooling power over three days shows an average cooling power > 110 W/m2 and a noon-time cooling power of 93 W/m2 between 11am – 2pm. The average nighttime cooling power is higher than that of the daytime, and the cooling power peaks after sunrise and before sunset. The measurement error of the radiative cooling power is well within 10 W/m2 (32)."
Fans won't drop temperatures below ambient. This will.