The easier way to precipitate chloride is using silver, you mix the salted water with silver nitrate and the silver chloride goes to the bottom. But you now have a sodium nitrate solution. (It's safe to do this experiment at home, if you are careful and follow minimal laboratory safetely procedures. But don't drink the water!) https://en.wikipedia.org/wiki/Silver_chloride#Chemistry
Just assume that all the sodium salts are soluble. The only example that I remember is the sodium uranil acetate salt. But it's definitely not a good idea to drink that water later. https://en.wikipedia.org/wiki/Uranyl_acetate#Safety
[1] I think that this is the main reasons why most of the salt in the sea is sodium chloride. Related question: Why doesn't the sea have too much potassium?
To answer your question, ion exchangers do what you ask, using sulfuric acid and sodium hydroxide, and produce a ton of sodium sulfate in the resulting desalinated water.
It's surprising there isn't some chemical reaction to precipitate out the sodium and chloride atoms bonded to something else, or whatever.