I just actually submitted a support ticket with Sapphire because I've been troubleshooting this for months and have completely exhausted every possiblity I could think of. I'm posting this here so that other people can contribute their experiences or suggest a fix.
Graphics: Sapphire 7850 1GB OC edition
CPU: Intel Core i3-3220
Motherboard: ASRock Z75 Pro3
PSU: Corsair 550VX
OS: Windows 7 64bit
Here's the text of the ticket I submitted:
I am using two monitors in cloned desktop mode. One monitor is an Acer P235H connected to the card through DVI using a brand new DVI-D cable. The other is a Samsung LN40C550 HDTV connected to the card through HDMI using a brand new HDMI cable. The resolution is 1080p60Hz (on both monitors of course since they are cloned).
The issue I am experiencing is that on one of the two monitors (and, as I will explain, it can be one or the other, but only one at a time) I see severe tearing/splitting. This is in the form of a single horizontal tear line across the screen, seen in all applications including games and the windows desktop environment, and noticeable at all times where there is motion and vertical lines (that makes the split especially visible). The tear line slowly "travels" up the screen. This effect happens whether or not V-sync is enabled, either in games or in the Catalyst control panel. This effect happens regardless of which dual display mode I am using (cloned, extended etc).
I cannot take screenshots to demonstrate this because this is clearly a refresh rate issue which cannot be captured by screenshot. I have instead taken a photo of my screen with a camera as I simply dragged a window around the screen. I have pointed out the (already very obvious) tear/split line with red arrows at points where there are vertical lines or other elements that make it especially obvious, but the line runs horizontally all the way across the screen. Hopefully this, along with my description, is adequate.
If I physically disconnect one of the two monitors, or disable one monitor through the Catalyst panel, the tearing completely disappears and the display is totally normal. This shows that this is an issue of dual displays and the two interfaces being slightly out of sync. Amazingly, I can switch which monitor shows the tearing (as I said only one of the two monitors is tearing at any one time). If I connect the Acer(DVI) first and the Samsung(HDMI) second, the Acer(DVI) shows tearing. If I connect the Samsung(HDMI) first and the Acer(DVI) second, the Samsung(HDMI) shows tearing instead! This shows that my monitors are not the source of the problem, that the issue is independent of connection interface, and that the card causes tearing in whatever monitor is connected second in order. I have the effect on every 13.x driver (including betas) released thus far (I have totally removed each driver using Driver Sweeper in safe mode before each new installation).
Before I bought this graphics card, I had a HIS Radeon HD 4870 installed with the exact same monitor setup, and did not experience tearing. Nothing else changed in my hardware configuration since I took that card out and installed the Sapphire 7850. At this point I cannot tell whether this is a faulty card (possible) or a design fault (more likely in my view) as I have seen other people report the same issue online.