  Faraday tomography of polarimetric observations at low frequency is a unique
tool to study the structure of the magneto-ionic interstellar medium (ISM)
based on Faraday depth. LOFAR data below 200 MHz revealed a plethora of
features in polarization, whose origin remains unknown. Previous studies
highlighted the remarkable association of such features to tracers of the
magnetized-neutral ISM, as interstellar dust and atomic hydrogen (HI). However,
the physical conditions responsible for the correlation between magneto-ionic
and neutral media are yet to be clarified. In this letter we investigate
further the correlation between LOFAR data and HI observations at 21cm from the
Effelsberg-Bonn HI Survey (EBHIS). We focus on the multiphase properties of the
HI gas. We present the first statistical study on the morphological correlation
between LOFAR tomographic data and the cold (CNM), luke-warm (LNM), and warm
(WNM) HI phases, separately. We use the Regularized Optimization for
Hyper-Spectral Analysis (ROHSA) approach to decompose the HI phases based on
the Gaussian decomposition of the HI spectra. In at least two fields of view --
Fields 3C196 and A -- out of four -- Fields B and C -- we find a significant
correlation between LOFAR and EBHIS data using the Histograms of Oriented
Gradients (HOG). The absence of correlation in Fields B and C is caused by low
signal-to-noise ratio in polarization. The observed HOG correlation in Fields
3C196 and A is associated with all HI phases and it is surprisingly dominant in
the CNM and LNM phases. We discuss possible mechanisms that would explain the
correlation between CNM, LNM, and WNM, with polarized emission at Faraday
depths up to 10 rad m$^{-2}$. Our results show how the complex structure of the
ionic medium seen by LOFAR data is tightly related to phase transition in the
diffuse and magnetized neutral ISM traced by HI spectroscopic data.
