Purely speaking my opinion:
The short answer...I see it possible to be an anarchist and a christian.

However, with the goal of anarchism being 'freedom and self-responsibility' I can see how some might defer pieces of these to the religious organization. This is where I see the split between spirituality (self reflection, emotional connection and feelings, etc.) and religion (the methodology/organization used to achieve spirituality.) I'm by no means bashing religion here, I just see these and distinct.

So after my long answer...being an anarchist and a spiritual person (who happens to use christian vernacular) I absolutely see no issue.
Being an anarchist and a christian, I have to say it depends on that individuals relationship to the church.

Things being dogmatically accepted without further thought, seem just as problematic as being indoctrinated by a government that frames information and produces propaganda. (i.e. Things are believed simply because they said so.)

Like everything else, it's about personal responsibility and freedom, which requires access to information and analysis.

Just my take :)