Relative trisections of fiber bundles over the circle

For an oriented $4$--dimensional fiber bundle over $S^{1}$, we build a relative trisection from a sutured Heegaard splitting of the fiber. We provide an algorithm to explicitly construct the associated relative trisection diagram, from a sutured Heegaard diagram of the fiber. As an application, we glue our relative trisection diagrams with existing diagrams to recover trisected closed fiber bundles over $S^1$ and trisected spun manifolds, and to provide trisections for $4$--dimensional open-books.