As for the Anglo-Papalists, once somebody realizes the Catholic Church is true, I am highly skeptical of them staying outside the Church for any reason. Corporate reunion? Please. A love of Chardonnay, canapés, and Cranmerian English are not reasons to stay outside the Church.The history of relations between Catholics (Catholicism) and Anglo-Catholics makes a good academic paper. Were Anglo-Papalists extreme Anglo-Catholics, what the English Protestant mainstream suspected all along about A-Cism, or the opposite of Anglo-Catholics? Anglo-Papalists were a tiny minority in both England and America, as Anglo-Catholics were, but they defined A-Cism in England last century.
Anglo-Catholicism at its start, like the high churchmanship that launched it, was NOT about wanting to come into the church (as we believe Catholicism is) but a RIVAL true-church claim, claiming a divine foundation and authority, and even a spiritual independence from the state, just like we do, but for King Henry's and Queen Elizabeth's "church," to OPPOSE us as well as the dissenting (non-Anglican, "free-church") Protestants such as Congregationalists (the extreme Calvinists who walked out of Anglicanism) and the Methodists (well-meaning John Wesley's movement, which likewise left Anglicanism). The A-C movement started in 1833 not at all a movement to copy our traditional ceremonial and bring England back into the fold; rather, it started as a protest against the British government acknowledging an effect of Catholic emancipation by considering suppressing four dioceses of the Anglican Church of Ireland, which of course next to no Irish belong to.
The 19th-century high churchman thought his church kept and would always keep the conservative, patristic line as he understood it, while the Pope had overstepped his bounds and was setting the dangerous precedent of innovating in doctrine. (The Pope having the final say in a diocese, Mary being all-holy, and the Pope sharing in the church's infallibility of course aren't innovations.) It would be fair to say they could well imagine the 21st-century Catholic Church ordaining women and having same-sex marriages while the godly Anglican Church stayed the course.
The Western world's scholars read each other's books and wrote to each other, so some high-church Anglicans knew a number of Catholics, yet there was also a belief that to be publicly too close to Catholics was disloyal, giving into a church they still believed was in error. Yet F.G. Lee's biographer writes that as early as the mid-1800s there were Anglo-Catholics of their time (not outwardly would-be Catholics) quietly doing exactly what the Protestant English hated and feared, writing among themselves and to friendly Catholics about a "reconciliation with honor with Rome" as some would call it now.
By the end of the 1800s, men such as Lee (even though Lee by then was an outlier, an "oddball," withdrawn from A-Cism, arguably having become its opposite, convinced of Catholicism) had changed the movement to what in England it often was and, among Episcopalians, what it was often taken to be: would-be Catholics imitating our rite and trying to steer Anglicanism into Catholicism. Betrayers of Cranmer et al.? You bet. Cranmer et al. were heretics. (American Anglo-Catholics copied our style too but believed in what they thought was Anglicanism; Hooker minus the Erastianism.)
The church says if you become convinced of Catholicism you must convert, but God is patient.
Reactions from Catholics were either privately supportive, encouraging this movement toward the church, or attacking those who "ape the true faith." I imagine the knowledgeable among the latter Catholics remembered the high churchmen's original intent, a rival true church still Protestant. (The English Church as the only lawful church in the realm, and some high-church Episcopalians believed the same in America; "the Roman Church" being "the Italian Mission to the Irish.") The former included Cardinal Mercier and the informal Malines Conversations in the 1920s at Anglo-Catholicism's height. (I guess Viscount Halifax was an Anglo-Papalist.) For naught because the A-Cs never spoke officially for Anglicanism nor for all Anglicans, many of whom are liberals/Modernists or Evangelicals.
You had the confusing situation when I was born into the Anglican scene. A-Cs had high-churched Episcopalianism in enough places that you could learn or infer a kind of pre-Vatican II Catholic spirituality from it. Vatican II had both the effect of encouraging would-be Catholics thanks to then very fashionable ecumenism, which was also why some Episcopalians who weren't really A-C outwardly high-churched themselves ("high church" originally referred to a high view of church authority, not high ceremonial), AND protestantizing/low-churching Catholics throughout the Western world so, thanks to Episcopal semi-congregationalism, Anglo-Catholic parishes sometimes remained conservative and pre-Vatican II vs. the local Catholic liberals/Modernists. Yet one could grab onto the hope that thanks to ecumenism, if we weren't already Catholics (which a number of Anglicans believed), we were on our way home to the church.
Of course Articles XIX and XXI gave away the ending centuries ago so history went in another direction from what we imagined.