The "Saving Private Ryan" star portrays American Civil War veteran Captain Jefferson Kyle Kidd in the western, in which he travels from town to town delivering readings of the national news.
However, the manner in which Hanks' character delivers the headlines to locals was a real sticking point for the actor and Greengrass because they didn't see eye-to-eye.
"We were halfway through the shooting day, and we ended up sitting on the wooden sidewalk of one of our Western towns. And we went at it, he and I, about what these performances of readings of the news meant," Hanks recalled to CinemaBlend.com.
"I was hell bent on authenticity and the real news and the real stories, and almost a dry perspective and presentation of them. And he was bent on the connecting of the audience, of the inspiring of the audience, of the reading of the audience and of the enthrallment of the audience with what this news was."
Hanks continued, "He was saying, 'You must understand: you are putting on a show to bring people together.' And I was saying, 'Paul, you've got to understand: I am reading the news to bring people together.' And we had to find this thing that was in there."
The double Oscar winner doesn't reveal who got their way, but the clash of words didn't dampen Greengrass' experience of working on the set of the 1870s survival drama, which is based on the 2016 novel of the same name by Paulette Jiles.
Greengrass, who previously worked with Hanks on 2013's "Captain Phillips", said, "It is a dream come true for me. In my many years as a filmmaker, I've done tough films about where we are today; I've done entertaining films about spies on the run, all sorts of things, but I had never made a classical, beautiful Western. That is what I've tried to do here."