The issue of poverty as espoused by the Franciscan order was a hotly debated theological issue in the 13th and 14th centuries. Eventually the Pope had to get involved to denounce the idea of "Absolute poverty of Christ" as heresy. The Franciscan revolution may have not gone as far as some people have hoped, but it was an important step in the development of Western liberalism (cf. Larry Siedentop's Inventing the Individual, pp. 288-292). Incidentally, this same debate was also featured as the central plot conflict in Umberto Eco's historical fiction "The Name of the Rose".
Quiz: who said, “all politics is, at bottom, theological?” St. John Henry Newman, or Proudhon? Trick question! Both! This is radically different from the Marxian/Nietzschian maxim, that all theology is, at bottom, politics. I don’t think anarchism is actually a complete, proper atheism; it tends to propose itself to a kind of negative theology (no false gods). Therefore dialogue is possible, unlike the Marxist, Nietzschian case, where atheism is complete and postulatory.
One area of commonality is that we both believe that society is natural. I’ve already mentioned Book II of Suarez’ Against the Errors of the Anglican Sect in this thread. Anarchists who read it will find it, in turns, dazzlingly profound and infuriating, yet astonishingly satisfying as a whole (just as I do with Graeber).
A second area of commonality is that we both believe in a horizon beyond politics. Graeber is especially good about this. For him, anarchism is an interior disposition — a mode of being — prior to any political program. He therefore avoids the infallible symptom of political atheism, the absolutization of politics. His books are about human life, and in particular about how politics can take place within life, rather than life taking place within politics. The intellectually fearless anarchist can read Andrew Willard Jones’ Before Church and State, and see the commonalities.
As for substantive content of dialogue, the anarchist can help the Catholic to see instances where the Church has, in practice, confused the eternal, Divine order for a particular, temporal order (see Pope Benedict XVI’s discourse on the temptations of Christ and the Kingdom of God in Jesus of Nazareth, Volune I for how to approach this with orthodoxy and openness). This will help the Church more authentically pursue its mission.
In turn, the Catholic can help the anarchist to resolve the power-authority distinction, which lies at the heart of the anarchists’ inability to sustain any positive ideal (see various 19th Century failures, Occupy Wall Street, and the more recent corporate co-optation of radical culture). From there, it is not hard to offer the highest authority, the Lord who stands in judgement of all Lords; sin as the universal oppressor, and oneness in the body of Christ as true freedom.
Some other texts I think we would all like:
St Cyprian of Carthage: On Almsgiving
St. Thomas Aquinas on the Universal Destination of Goods (Summa Theologica)
St. Thomas Aquinas on Distributive Justice (Summa Theologica)
Pope Francis’ Laudito Si
Anything by or about Bl. Oscar Romero
St. Augustine’s City of God
Most of Pope Benedict XVI’s scholarly work, Jesus of Nazareth already mentioned as a highlight.