ROME — Pope Francis dropped in again this week on his predecessor, Pope emeritus Benedict XVI, wishing him happy birthday “with particular affection” in a now familiar showing of white-cloaked cordiality.

But behind the friendly visit, the talk of conspiracies and competing power centers is swirling inside the Vatican and far beyond. Just last week, Benedict, who turned 92 on Tuesday, released a 6,000-word letter holding forth on his views on the origins of the Roman Catholic Church’s clerical sex abuse crisis — effectively undercutting Francis on a contentious issue that has roiled his papacy.

For many church experts, the letter marked the most recent, and egregious, example of why having two popes — whose homes are separated by a few hundred meters but whose style, substance and visions of the church are vastly apart — can be so confusing to the faithful.

To be clear, Francis is the pope and is in charge. He is the one who can promulgate dogma and whose papal pronouncements when speaking “ex cathedra” — with the authority of the office — on questions of faith and morals are considered infallible. Benedict gave all that up — including the infallibility — when he stepped down.