I'm using one of those faucet mount Pure filter thingy. The water tastes absolutely devoid of any "flavors." I lack any ability to make judgement beyond that. I use water from it to make stock/working solutions.
It is said to remove 99.xxx% of mineral content.
I have a whole house filter set-up that does the exact same thing your tap filter does, except at every faucet and plumbing fixture that draws cold water. I filter for sediment and taste, two different filters in line on the main feed line to the house.
I have used RO for 10 years because our water source is well water. I found RO is very economical for houses with well water. The only consumable is water and well water is almost free except the electricity used by the well pump.
Local tap water is usually around 120-180 ppm. The water come out of RO is around 10 ppm. Sometime I feel the RO water is too pure and might not have enough minerals to satisfy human needs.
Home distilled water costs around 10 cents or 20 cents per gallon. This is cheaper than buying from the store.
Both distilled water and RO water are perfect for darkroom uses.
I noted only one comment mentioning a de-ionizer. My mother was a tech in a large hospital lab that ran a de-ionizing column 24/7. It requires a modest amount of electricity, and provided enough water for all the labs, and several photographers who filled up every week or so. The one in the lab sat on top of a cabinet, a footprint of only a couple square feet. I don't know if a de-ionizing system would meet your needs, and I don't know what kind of maintenance or expendables it required (I just filled a couple of 1 gallon jugs a couple times a week or so, and there were several photographers that I knew of in the lab taking home water also). It might even be the kind of thing that can be owner built?
"We often think that when we have completed our study of one we know all about two, because 'two' is 'one and one'. We forget that we have still to make a study of 'and'."
-A. S. Eddington
As far as electric bill..
Well, there are solar distillers if You want to cut costs. I have friend who made one for his house, since its relatively easy to be made by someone who knows the first things about the process.
@bwfans - the human body gets minerals from the food by hydrolysis, also some of the companies that sells distillers, provide liquid minerals, so You know what You put in the water and how much.
Some of us drink distilled or RO, their entire life.
As far as I know distilled water is what is provided in the military ships, submarines etc. etc.
For fussy work like color separation negatives or matching masks where the development characteristic
must be exact and perfectly repeatable, or making up critical pH dyes, I only used steam-distilled. It genuinely makes a difference.
Not an answer to the question, but, anyhow, I used the collected water from my central AC unit for years until warned about possible fungus growth. * then began boiling it for about five minutes before filtering. I now have a UV unit installed in the AC condensor unit that kills possible fungi. I have no problems; the water is free; and my houseplants love it, too.