Flashing the card to the unlocked MSI BIOS worked quite well for me. I have full control of my card (core/mem/voltage/fanspeed) right from Afterburner. By the way, the HD5850's stock voltage is 1.088v, which is actually a few notches lower than the HD5870's 1.150. Since they have the same cooling setup, I don't see any harm in setting the HD5850 to 1.150v as well. I was able to attain core 875MHz / mem 1175MHz at that voltage without overriding the BIOS fan controller.

/EDIT: for reference, I have a Gigabyte (and soon a Sapphire as well) card. Many people have probably noticed though, the only difference between the cards is the decal stuck to the cooler to differentiate from other companies. The complete package is still being manufactured in AMD's own facilities, so in all honesty, any BIOS should work with any card at this point in time.

This may change in the future as the companies start to buy just the core GPU chips and implement their own PCB/cooler setups, but for the time being my belief is that they're all 100% interchangeable.
/EDIT