Sometimes, the phosphates, when tested, will show up as "zero" but that would be due to the fact that the hair algae that you have is consuming what phosphates are available to it. If you get a "zero" reading after testing with your new kit and have no obvious explanation for the hair algae since your nitrates are showing 0ppm, then it's most likely the situation I mentioned. I don't believe you should have to resort to RO/DI water unless, when you test your tap water, it shows some elevated nitrates and or phosphates. If it does not, then address whatever it is right there in the reef that's making nutrients available to the hair algae. The measures are basically:
-reduce amounts of feedings of fish and coral
-increase the amount and frequency of water changes
-make sure your skimmer is operating efficiently or get a more effective one
-add a sump with a refugium to your setup
-vacuum your substrate
It wouldn't hurt to add a Phosphate Reactor if your efforts don't seem to be making a dent in the hair algae. Using a good quality medium like Rowaphos is effective.