I don't think no processed food items is a requirement of clean eating. I believe that the bulk of your foods should come from unprocessed sources. Personally, I think that there's nothing wrong with having a couple slices of pizza, ice cream or something of that sort as long as the rest of your diet is in check.
In my honest opinion, most forum people who actually make an attempt to eat clean go too far and don't allow for these treats, when it wouldn't really matter one way or the other as long as everything else is in check, and consumption is limited.
The Greatest Respect You Can Earn is Self Respect.