I've had the same exact problem with my guitar's Floyd-Rose setup.

They're a lot of trouble to get set up, but once you get it in tune, it's worth it. I don't think I've tuned my guitar in a month, and it always stays perfectly in tune, even when I'm doing all sorts of crazy whammy stuff and bending like a mofo.

Jarviss gives some good advice. Since the bridge is "floating" on the springs, the tension of each string affects the tension of the others. This holds especially true when you're tuning the lower strings, which are fatter and hold more tension. When you tune the lower strings up, the higher strings will go flat. And vice-versa.

To get around this you can try one of two things.

You can first tune all the strings using your tuner. Your lower strings will end up being flat and the higher strings will be sharp. Then raise your lower strings by a half step or so, and tune everything relatively using harmonics. Everything should fall into place, but you will have to repeat this process almost every time you play because the tension across the bridge is unbalanced.

Or you can just tune your guitar with the tuner, (the low strings will be flat and the high strings sharp again) and just tune everything relatively from the low E string. After doing this a few times the guitar will find its natural place where the tensions are even across the bridge, and everything will be cool. This is what I do, and the guitar usually ends up somewhere between E and Eb.