One of Berkey's selling points back in the day was that it left some beneficial minerals in your water. I don't know what they say about it now. The trace amount of minerals is what makes it taste so good IMHO. Our water has a softness to it after it goes through the filter...only way I can describe it. Everyone loves it.We have hard water here. Berkey does not remove all of the stuff. But it does remove a lot. I have run a TDS on before and after, and gotten lower TDS after. Mainly because I was wondering if it would work to replace distilled water. And that answer would be - nope. Not meant for it, can't do it.
Btw, there is a filter that says it will do that. It's called Zero if I remember correctly.
Again, Berkey does not say it will make distilled water. Or that it will occur naturally occur minerals that are below a certain size. None of the hiking/emergency use filters offer that promise, Not Katadyn (or the many variants they sell), not LifeStraw, not Sawyer. If that is what you are looking for, you need to be looking at different systems. Heck, Sawyer doesn't even promise good tasting water, just safe from certain things. (And reports are that if the water has a nasty natural flavor beforehand, it will still have it afterwards.)
You can't buy apples and expect them to suddenly become oranges. Well, you can, but then you'd be as delusional as some of the rest of the nuts out there.
I don't like the taste of distilled water and I've heard if that is the only liquid you consume, you would eventually be in trouble.