  The atomic interstellar medium (ISM) is observed to be full of linear
structures that are referred to as "fibers". Fibers exhibit similar properties
to linear structures found in molecular clouds known as striations. Suggestive
of a similar formation mechanism, both striations and fibers appear to be
ordered, quasi-periodic, and well-aligned with the magnetic field. The
prevailing formation mechanism for striations involves the excitation of fast
magnetosonic waves. Based on this theoretical model, and through a combination
of velocity centroids and column density maps, Tritsis et al. (2018) developed
a method for estimating the plane-of-sky (POS) magnetic field from molecular
cloud striations. We apply this method in two H\textsc{I} clouds with fibers
along the same line-of-sight (LOS) towards the ultra-high-energy cosmic-ray
(UHECR) hotspot, at the boundaries of Ursa Major. For the cloud located closer
to Earth, where Zeeman observations from the literature were also available, we
find general agreement in the distributions of the LOS and POS components of
the magnetic field. We find relatively large values for the total magnetic
field (ranging from $\sim$$\rm{10}$ to $\sim$$\rm{20} ~\rm{\mu G}$) and an
average projection angle with respect to the LOS of $\sim$ 50$^\circ$. For the
cloud located further away, we also find a large value for the POS component of
the magnetic field of $15^{+8}_{-3}~\rm{\mu G}$. We discuss the potential of
our new magnetic-field tomography method for large-scale application. We
consider the implications of our findings for the accuracy of current
reconstructions of the Galactic magnetic field and on the propagation of UHECR
through the ISM.
