Those inconsistencies are exactly when historiography becomes crucial to historical understanding. Disagreement between sources raises questions about the quality of the sources and those who reported them. The risk of not addressing these issues is that erroneous information can become accepted, and must then be recast in light of the qualities of the sources.
For example, in the late 1800s when the historian H. H. Bancroft was working on his monumental Histories of California and Alaska, he employed a Russian-speaking researcher and translator named Ivan Petrov. This fellow discovered a number of archival documents about the Russian-American Company that Bancroft cited. Years passed and many other authors cited Bancroft. Much later, historians discovered that Petrov was the author of several fraudulent documents. Bancroft's credulity allowed his work to be tainted by clever fictions, and caused generations of confusion. Now, the only responsible means of using or understanding Bancroft's text is to incorporate the story of Petrov's deceit. Rather than ignoring or denying or emphasizing it, we simply regard it as a key aspect of the works it affected.
Sorry I have no citations, as I am not well-read in historiography.