I had a somewhat similar encounter with Pope John Paul II. I was just strolling through St. Peter's on a whim. He passed within 10 feet of me, the crowd's reaction was the same.
I did too. It was at a field Mass in Canada (BC) and we were filing back towards our bus which would take us back to Washington. My friend and I were impatient at having to walk so slow after sitting in the sun so long, and we stepped around a barricade to duck behind the altar so that we could walk faster than the crowd shuffling along in front of us. We ran smack into Pope John Paul II! We had no idea he was still there, a she had disappeared into the altar at the end of the Mass.
We were both so startled the only words we could muster were, "Thank you Father" followed by a bow.
He gave us a broad grin and a little wave, jumped into the Popemobile and was off in a cloud of dust!