...I don't carry it around with me when there. I do hide it in the hotel room, though, so it doesn't get taken. It's a good idea to take a couple photocopies of it and other important papers and put the copies in different bags, just in case your actual documents get stolen. Can make it easier (I'm told) to get a replacement passport at the consulate.

As a missionary in Europe we carried ours at all times, but I was only ever required to produce it once, in Belgium, at a random stop. Obviously you must produce it when crossing borders (maybe not from one EU country to another), but I'm guessing you'll have your bags with you when you do that.