I swear Blizzard touched upon this topic (as in, the consequences) back when its Real ID system was first mentioned, but apparently everyone forgot or I made the whole thing up. Essentially, its main purpose is to let players use real names when communicating with friends.
Word has gotten out -- straight from Blizzard, actually -- that, at least for the company's forums moving forward, the use of Real IDs is not optional.
Your real-life name will be what appears on posted messages, presumably to encourage users to not be complete b*stards knowing their actions could potentially be found by a future employer doing a preliminary history check on Google. Here's the quote of choice from Blizzard:
"In the near future, anyone posting or replying to a post on official Blizzard forums will be doing so using their Real ID ... 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."
As someone who, on a daily basis, has to deal with what Internet anonymity does to otherwise reasonable people, I find this whole ordeal to be kind of hilarious. But yes, the issue of privacy is a serious one; there's always the option to, y'know, not post on Battle.net if it's a major concern to you, though. I get the feeling many people will take that route.
[*].disqus.comto your security software's whitelist.