To be honest I haven't yet seen a study that indicates that it's actually unhealthy to drink distilled water if you're otherwise eating a healthy balanced diet. The only study that seems to indicate that it's a problem is if a person is already seriously malnourished and the trace amounts of minerals in water was actually making up a significant portion of their intake of those minerals.

I eat several cups of green vegetables a day which has a thousand times more of all of those minerals then tap water. I'm not concerned that missing out on a few micrograms of sodium, potassium, calcium, and magnesium is going to cause a problem if I'm otherwise getting all of those by the milligram in my food.

If you have studies to show otherwise I'd be happy to see them. I did buy a mineral supplement to put in water but it makes it taste pretty salty so I haven't been using it very much.