C'mon guys, this isn't rocket science. Keep it modular and keep it off-the-shelf, right?
First stage, convert a regular evaporative cooler to pump the dessicant solution instead of water. It just needs to be highly corrosive-resistant. Now your incoming 100F air at 40% RH is now like 120F at 10% humidity or something like that. Hotter, but drier.
Next, you need a heat exchanger to pull some of that energy out without adding humidity. Adobe Air makes a modular product to do this. Or, you can skip this piece and get a two-stage evap such as the OASys. Or just insert a Coolerado here.
Finally, to really get cool temperatures, we need to add water back in. If you inserted a two-stage unit above, you're done. Otherwise just use a regular-ole evaporative cooler here.
Since we actually want to make this feasible without vast quantities of free energy, we will recharge the dessicant in a more sustainable manner. Obtain two empty swimming pools. The first pool is for the dry dessicant, the 2nd pool for the wet dessicant. After the cooling season, use a small solar concentrating array to recharge the dessicant pool. It'll take all winter but that's fine.