Context : The local Galactic H i gas was found to contain cold neutral medium ( CNM ) filaments that are aligned with polarized dust emission .
These filaments appear to be dominated by the magnetic field and in this case turbulence is expected to show distinct anisotropies .

Aims : We use the Galactic Effelsberg–Bonn H i Survey ( EBHIS ) to derive 2D turbulence spectra for the H i distribution in direction to 3C 196 and two more comparison fields .

Methods : Prior to Fourier transform we apply a rotational symmetric 50 % Tukey window to apodize the data .
We derive average as well as position angle dependent power spectra .
Anisotropies in the power distribution are defined as the ratio of the spectral power in orthogonal directions .

Results : We find strong anisotropies .
For a narrow range in position angle , in direction perpendicular to the filaments and the magnetic field , the spectral power is on average more than an order of magnitude larger than parallel .
In the most extreme case the anisotropy reaches locally a factor of 130 .
Anisotropies increase on average with spatial frequency as predicted by Goldreich and Sridhar , at the same time the Kolmogorov spectral index remains almost unchanged .
The strongest anisotropies are observable for a narrow range in velocity and decay with a power law index close to –8/3 , almost identical to the average isotropic spectral index of -2.9 < \gamma < -2.6 .

Conclusions : H i filaments , associated with linear polarization structures in LOFAR observations in direction to 3C 196 , show turbulence spectra with marked anisotropies .
Decaying anisotropies appear to indicate that we witness an ongoing shock passing the H i and affecting the observed Faraday depth .