Daniel,
I won't claim to be any sort of an expert, especially since I also have an algae problem I have yet to conquer, but I'll put in my two cents.
First of all, you mention you don't want to lose the live rock. Unfortunately, turning off the lights completely will kill both the problem algae and the nice coralline algae on your live rock. It wouldn't hurt to cut it back by a few hours, but understand that taking away all light from the tank will kill your live rock.
Also, you say your RO water has no phosphates. Are you using a DI filter on your RO setup? If not, silicates could also contribute to the algae problem, at least they did in my experience. I'll assume you're using a high quality salt mix, so that rules out another source of the phosphate. So, the phosphates probably came from feeding. Your best bet to lower the phosphates would be routine large water changes.
Once the phosphate starts to lower (oh, and I'd also suggest a chemical phosphate remover like others have said) then reduce the water changes, since they add trace elements that the algae will like. Skim vigorously and soon the algae should start to diminish. It won't go away forever, but with light stocking, giving just enough light but not too much, and vigilant maintenance on a regular basis, your tank will be looking great again. Oh, and be sure to keep an eye on your cleanup crew and replace any snails or crabs that will die over time.
I give this advice because it's what has helped me the last few months. My 20 gal was great for a few months and then had a huge algae bloom. What I've suggested above is what has helped me. I upgraded my skimmer and focused on improving water quality. I bought a nice RODI unit to reduce the unwanted trace elements I had put in there by using chemically-treated tapwater for a few months. I stopped trying to use additives and such, becasue they just don't work well. Good aquariums are a result of hard work, not miracle chemical compounds. My algae is almost gone, and I'm setting up another small tank to take a few fish out of my 20 gallon to relieve the bio load a bit. If you'd like, I can post some pictures. I think I have pictures from during the worst of the bloom and again now that it's getting better.
Well, anyway, sorry this is so long, but algae control is never a simple topic!
Good luck, Daniel,
Joe S.
:happyfish