Tidal streams of disrupted clusters are powerful probes of the gravitational potential of the Galaxy and they are routinely detected in the stellar halo of the Milky Way. It was recently shown that tidal streams of open clusters can now also be detected within the Milky Way disc. In this work, we highlight the fact that disc tidal streams also provide a powerful new diagnostic of the non-axisymmetric disc potential and may, in principle, provide a new constraint on the pattern speed of the Galactic bar. In particular, we show how the stream-orbit misalignment for an open cluster on a quasi-circular disk orbit in the solar vicinity varies as a function of the position with respect to the bar resonances. The angular shift rises beyond corotation, reaching values as high as 30° close to the outer Lindblad resonance (OLR), then dropping again and reversing its sign beyond the OLR. We applied this mechanism to the recently detected tidal stream of the Hyades open cluster and we note that the detected stream stars would be very similar when taking a potential a priori with no bar or with a fast pattern speed of 55 km s−1 kpc−1 (or lower than 30 km s−1 kpc−1). However, we find that candidate stream stars are different than previously detected ones when adopting a potential with a bar pattern speed of 39 km s−1 kpc−1, which is consistent with the most recent determinations of the actual Galactic bar pattern speed. Previously detected Hyades candidate members would, on the other hand, favour a barless galaxy or a fast bar of pattern speed 55 km s−1 kpc−1. Interestingly, the previously reported asymmetry in star counts within the leading and trailing tails of the Hyades tidal stream persists in all cases. Our study conclusively demonstrates that the effect of disc non-axisymmetries cannot be neglected when searching for tidal streams of open clusters and that current candidate members of the Hyades stream should not be trusted beyond a distance of 200 pc from the cluster. Moreover, our study allows for ideal targets to be provided for high-resolution spectroscopy follow-ups, which will enable conclusive identifications of the Hyades stream track and provide novel independent constraints on the bar pattern speed in the Milky Way.