Ok, so the 348 mg/l of Ca2+ ions translates to about 869 mg/l of CaCO3, which is a lot.

If you really used water like that for washing clothes, all those Ca2+ will suck a bunch of fatty acids from the detergent, forming loads of soap scum. Technically, it's Ca and Mg salts of fatty acids, but it's nasty either way. Yes, the Mg ions won't sit idle. They will join the party.

Anyway, I don't think the picture is as crazy as it seems at first glance. If your local water is loaded with multivalent cations (Ca2+, Mg2+, Al3+ Fe3+ etc.) dumping in about quarter a bottle of detergent for a single batch might not be totally unreasonable after all. Gets pretty expensive though, but at least you can practice your detergent+softener art every time. Hobbies always cost something, and there are worse ways to spend your money. Personally, I still prefer latte art, but whatever floats your boat.

Oh, and the precipitates! All of that goo has to go somewhere. Some of it is going to get stuck on the clothes, inside the washing machine, pipes and every surface it comes in contact with. Cleaning up a mess like that should keep you busy.