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We present ALMA observations at a spatial resolution of 0.2 arcsec (60 pc) of CO emission from the Taffy galaxies (UGC 12914/5). The observations are compared with narrow-band Pa$α$, mid-IR, radio continuum and X-ray imaging, plus optical spectroscopy. The galaxies have undergone a recent head-on collision, creating a massive gaseous bridge which is known to be highly turbulent. The bridge contains a complex web of narrow molecular filaments and clumps. The majority of the filaments are devoid of star formation, and fall significantly below the Kennicutt-Schmidt relationship for normal galaxies, especially for the numerous regions undetected in Pa$α$ emission. Within the loosely connected filaments and clumps of gas we find regions of high velocity dispersion which appear gravitationally unbound for a wide range of likely values of $X_{\rm CO}$. Like the "Firecracker" region in the Antennae system, they would require extremely high external dynamical or thermal pressure to stop them dissipating rapidly on short crossing timescales of 2-5~Myrs. We suggest that the clouds may be transient structures within a highly turbulent multi-phase medium which is strongly suppressing star formation. Despite the overall turbulence in the system, stars seem to have formed in compact hotspots within a kpc-sized extragalactic HII region, where the molecular gas has a lower velocity dispersion than elsewhere, and shows evidence for a collision with an ionized gas cloud. Like the shocked gas in the Stephan's Quintet group, the conditions in the Taffy bridge shows how difficult it is to form stars within a turbulent, multi-phase, gas.