Hi Diana,
You're thinking of a water softener. These use a resin to exchange calcium and magnesium ions for sodium ions (the resin can be recharged by soaking in a salt solution). Deionized water uses a resin which exchanges hydrogen for other ions. R.O. water removes ions by forcing the water through a membrane. In my case, I use the DI water to remove excess phosphorus, since we already have soft water here. I have to add back in some amount of calcium, magnesium, and sodium even for the soft water systems.
However, I don't think that water softeners would create fish problems from the excess sodium. These would be the standard filter resin pillows that people use in discus tanks and the like. To remove only hardness, this is a far less expensive option. But it won't remove phosphorus.
I'll definitely keep you posted about the softwater tank! I'm just as curious as you to see what will happen.
-Laura | 
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