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While Christianity in the early medieval Celtic world did have a number of distinct traditions and practices, the idea that Celtic Christianity (also known as Insular Christianity) ever formed a distinct religious organization is generally rejected by mainstream historians. This is due in large part to a lack of evidence supporting the existence of a "Celtic Church".
But I don't think it necessarily had to be this way. So here's your challenge: have there be a distinct Celtic Church that's entirely separate from the "Roman" one.