I'd just use the naturally hard tap water. FWIW, I once lived with hard sulfur water and it made fantastic coffee. lol!

Thank you for that. I didn't mention my soil, because we're not growing directly in it (it's clay, pure clay in places, clay-loam in others). But I will do some testing on the compost-soil mix I use. Thanks for the tip.

Hard water isn't harmful, it just has a high mineral content. Softened water is not recommended for plants because it has more sodium.

Thank you, lgteacher. It's the sodium in softened water, not the chloride, that is the problem.

Compost tends to get more acidic so it would tend to counter your tap water a bit. You should get yourself a pH test kit and check your soil. One of the ways of reducing pH in soils is to sprinkle sulphur. But it's slow acting so there's no instant results.

Actually, I believe that acidic compost is an exaggeration.

https://www.gardenmyths.com/compost-creates-acidic-soil/

In any case, buffering action of most soils will knock down the alkalinity of irrigation water. My own soil is pH 7.9, but my tap water is 9.3. Egads! That comes from lake water which is 8.3, but the city softens it by adding lime. My limestone-based soil has no trouble buffering it to a more plant-amenable pH. But soil acidification would be very hard.

Wood ash is extremely alkaline. Of course if it's added to compost it would make that alkaline. So obviously ... don't add it. That's one of the reasons wood ash is recommended for compost, to counter acidity. Composts can be made to have lower pH, choose your ingredients. The other thing to note is that organic methods are slow, that's why I said "Compost tends...". Elemental sulphur is an accepted organic soil additive. It too is slow acting, but it works. It's popular with tropical fruit growers as those, in general, like lower pH. Pineapples grow well in pH 5.5 soil.

Well, again, if you have high pH water, acidification with sulfur is going to be highly temporary, if it works at all. Alkaline soil irrigated with highly alkaline water is never going to be acidic. To me, wood ash is toxic waste. Period.

Interesting discussion. Thanks all!