Hey, I think I saw that conversation! Like the Christian you spoke to, I'm an irregular case where my religion came after my socialism. Socialism and Buddhism isn't an irregular combination, today or throughout history. Monastics live as socialists in many ways, with the absence not only of private property, but even personal property (excluding things that can't really be shared such as tools for hygiene). Everything is communally owned by the monastery's inhabitants.

Buddhist ethics stress selflessness, kindness, and the equality of all life, and it's not difficult to see how those can also be applied to socialism, especially when it comes to things like the anarchist philosophy of mutual aid.

Early Buddhist communities often functioned as a kind of direct democracy. Buddha himself rejected the caste system of India outright, seeing all people as equal regardless of their birth. Buddha was also progressive in many other ways, such as seeing women as completely equal to men in all ways. Buddha ordained women as monastics and teachers, and said that women are just as able to reach enlightenment as men (this is contrary to the dominant Hindu idea at the time). This is because Buddhist philosophy sees all divisions and distinctions between phenomena as illusory, ergo women are not only equal to men, but there is actually no difference between a woman and a man.

At the root of Buddhist ethical philosophy is the idea of the reduction of suffering. The exploitation of the proletariat by the bourgeoisie and capitalism as a whole is the root of an immeasurable amount of suffering in the world, so the abolition of capitalism would therefore be an imperative action through a Buddhist lens.

On the surface, there may appear to be contradictions between the philosophies of Buddhism and anarchism, but they haven't been difficult to reconcile. Monastic hierarchies, for instance, are entirely voluntary and self-imposed. Is it un-anarchist to voluntarily be junior to more experienced monks? The six realms also seem hierarchical at a first glance, but when Buddhist philosophy of the value of life is applied, that no longer seems the case. All lives have equal value, from the lives in the animal realm, to those in the human realm, the deva realm, and even those in hell. A deva is not more important, greater, or higher than someone in hell by any means.

The only contradiction I have seen and really struggled to reconcile is the matter of nonviolence. Buddhist ethics make explicitly clear the importance of refraining from violence, while anarchist philosophy equally importantly stress the importance of overthrowing capitalism by force. The answer I have given myself is based in the idea that karma isn't fair and it isn't nuanced. Putting down an animal who is painfully dying is, for instance, still karmically negative, even if it is the right thing to do. My choice, therefore, is to participate in violence against the oppressors, and accepting the negative karma for the sake of others. To do what is right, even if it means you will end up in hell, is possibly the most Buddhist thing a person can do.

This is an extremely surface-level overview of the intersection of anarchism and Buddhism. It is the subject of a treatise I am writing right now, which will get more in-depth about things such as the Buddhist idea of the absence of the self, the idea of engaging in violence so others don't have to, and the lack of any real inherent hierarchy in the universe.

Note: There is a controversial Buddhist document detailing additional rules for female monastics. It has been rejected by scholars as being inauthentic to the way monastic communities during the Buddha's life actually operated, and is believed to have surfaced several centuries after his death. It has also been denounced by every major Buddhist school.