They use clay beads, which are an aluminium mesh with H and OH groups weakly bonded to it (a weak ionic bond like this is called a "Ligand bond"[1]).
Firstly, it filters out bacteria and "bits" because of the porosity of the mesh (works like pushing current through a complicated network).
However, additionally it filters out non-H-or-OH ions from the water, which have a stronger Ligand bond with the Al mesh. (Obviously you eventually run out of H and OH ions in the network, as they're all replaced, which is when you need to change your filter.)
As such, we used clay-filtered tap water as 'distilled' water in all our experiments, because the worst it contains is bacteria and dissolved salts. I can't speak for the Coca Cola though, that's weird. I'm sure organic molecules would be small enough to get between the beads but too heavy/not ionic enough to bond to the mesh, or something.