Starting on July 27, according to a blue post on the Battle.net forums,
"Anyone posting or replying to a post on official Blizzard forums will be doing so using their Real ID - that is, their real-life first and last name - with the option to also display the name of their primary in-game character alongside it. These changes will go into effect on all StarCraft II forums with the launch of the new community site prior to the July 27 release of the game, with the World of Warcraft site and forums following suit near the launch of Cataclysm. Certain classic forums, including the classic Battle.net forums, will remain unchanged."
The post mentioned incidents of trolling, flame wars, and other low-content posts as reasons for the change. It's tough to see how this will pan out, as Blizzard's forums don't exactly have the best reputation, and wiping away the anonymity may just shame flamers into intelligent discussion. The Internet being the Internet, however means that people can still make fake names and accounts, so it may simply end up being people using their real names and posting like they did before, and people using fake names to continue trolling.
Source: Rock Paper Shotgun