Two decades before the Second Vatican Council, a section of the Catholic Church decided to enact sweeping reforms of its own. The Brazilian Catholic Apostolic Church, or I.C.A.B., abolished latin, celibacy and confession, embraced religious freedom, and redefined the pope as just another bishop. I.C.A.B. became the renegade church within a church. With over half a million members in Brazil, and up to four million in a complex worldwide network,...