Well, here is some info you might be able to use when looking at your card. Apologies as this applies to a 5870, however it looks like the same voltage controller can be used on some more recent cards, like the Nvidia 560.
Even with worst case scenario, here is how easy it is to do a pencil mod on a non-reference bare bones basic 5870 that completely and utterly lacks all voltage control. So, even if your 7850 is in the same worst-case scenario, I bet there is a similar opportunity for "tuning" the voltage using a pencil. The link is Italian, but Chrome does a pretty good job translating. Main point is to show what the voltage controller looks like on the 5870 and where to try looking on your 7850, on the backside of the circuit board, probably near the power connector (you won't have to take off the cooler - see the lower left area highlighted as the zona d'intresse della pencil mod). At the time, people bad-mouthed this 5870 on the reviews about how it couldn't overvolt, so its price dropped and I bought one and used this method:
http://www.hwupgrade.it/forum/showthread.php?t=2187874
As you can see, the voltage controller is quite tiny, about the size of a screw hole and its surrounding grounding metal. This low-priced voltage controller may be marked as uP6213. Anyway, a couple pictures down the italian site shows where to draw a pencil mark between the yellow and green spots. Perhaps there is an identical spot on your card to do the same. I've done this mod on a 5830 with very stable and predictable results, but I also used a volt meter to check the resistance of the pencil lead, then draw a little more, recheck, etc. Note that the picture is really blown up huge, the space to draw is thinner than most no. 2 penci leads and I had to use a mechanical pencil for good control.
Here is another site showing the MSI 560 with the same uP6213 voltage controller:
http://www.techpowerup.com/reviews/MSI/GTX_560_Twin_Frozr_II_OC/4.html
But also note, even with the same voltage controller chip, the 560 was able to implement a very basic driver control for software voltage control. The 560 site explains how even though the chip doesn't speak the common 'language' of chip communication (I2C, see
http://en.wikipedia.org/wiki/I%C2%B2C), it still can work based on a limited range VID based API.
So there is still hope, as I bet the 560 shows the natural evolution that video card makers have followed when using this basic chip, where back in the 5870 days you had to use a pencil mod, but with the 560 there is more control.
Remember too that you are a trailblazer by getting such a new card, the software guys still have to catch up and reverse engineer how the heck to get voltage control working for your card, at least I have hope for that.