Given what a stickler Apple (AAPL) is about protecting its trademarks (viz. its battle with Amazon over the phrase “app store”) you might think its lawyers would have done a thorough search on the term “iCloud” before Steve Jobs started waving it around like a banner for the post-PC future.

If they did, they failed to come to terms with a little Phoenix-based firm called iCloud Communications that claims it’s been using that name for its VoIP (voice over Internet protocol) service business since 2005. Because on Thursday, even as more than 5,200 developers were learning how to write software for Apple’s iCloud in San Francisco, the company was hit with a trademark infringement suit filed on behalf of iCloud Communications in U.S. District Court, Arizona District.

It’s a curious situation. Apple applied for an iCloud trademark in Europe on March 31, the same day it issued the first press release that used the term. At the time, according to a trademark search conducted by CNET, there were only two records of iCloud trademarks in the U.S. One was registered to a Swedish company called Xcerion that reportedly sold iCloud.com to Apple for $4.5 million in April. The other was filed on May 6 by a North Carolina man named Douglas Dane Baker.

A search on trademarkia.com Friday turned up 11 iCloud applications, all filed by Apple on June 1. It  did not turn up any filings by iCloud Communications.

Via The Next Web, which has made the text of the complaint available as a Scribd file here.