As far as I can tell, salt precipitation in the Dead Sea is a result of evaporation. Water can only hold so much salt in it. If there’s too much, the excess gets pushed out into the solid phase as salt crystals. It’s all about the solubility of each compound, which depends on all sorts of things such as temperature, pressure, pH, other ions, etc. As the conditions change, solubility changes, excess salts get precipitated and the solution finds a new equilibrium.