MIT had a spin-out company some years ago doing HDH (Humidification-Dehumidification) desalination.
In thermal cycles, the problem has been in the condensation step. If there is a carrier gas present this inhibits heat/mass transfer at the condenser surfaces. The usual way of getting around this has been to operate the system with no carrier gas, but that requires pressures below atmospheric pressure, requiring strong walls to withstand external atmospheric pressure.
The MIT invention was a bubble tray contactor, where air is bubbled up through trays of progressively cooler water. The water/air bubble interface provides a large surface area at low cost. One of the markets for this was cleaning up brine from fracked wells.
The company, Gradiant, is still around but has evolved to involve a wider range of water treatment technologies (which is very sensible from a business viewpoint, as customers buy complete solutions, not individual technologies). https://www.gradiant.com/