I'm going to share what I was taught by my Native American Anthro professor.
Many Amerind tribes practiced "reciprocity." This socio-economic system is much like communism: the community shares with one another, does for one another, and provides for one another, as each member would care for him- or herself. This is often thought of as "The Golden Rule," but it's at once more complicated and simpler than that.
For instance, among the La- Da- Nakota (three branches of basically the same peoples) was a concept known as mitakuye oyasin (mee-TAH-ku-yay oh-YAH-sin) or "all my relations." In this concept, everything is related to you, and you to it. Not just part of your family, but part of you.
When you share with another, there is no sense of obligation or repayment. You help your neighbor eat, because you are feeding yourself, and so on. This is a spiritual kind of connection, but it is important to note it is NOT RELIGIOUS in any sense. The pre-European La- Da- Nakota had no religion in the sense we tend to think of it. The Great Spirit you hear about so much wasn't a single god, but rather the mysterious whole of the spirits of everything. Everything is wakan, mystery. You are just as much the Great Spirit as a rock, or a bit of dung, or a baby, or a tree.
My professor taught us that Soviet Communism was modeled on the Amerind system, but left that spiritual part out. You can't instantly create a reciprocal system; it develops slowly. You can't just say, "Okay, now we're all going to share, and like it!"
However, if you have been born into a social system that tells you from day-one that everything is related to you, is a part of you, and must be cared for as you care for yourself, then reciprocity becomes the natural embodiment of that philosophy.
Christianity is too individualistic to function in that model.
Thus, Communism would have been no more successful with Christianity as a factor than without it.