The interaction of an extreme-ultraviolet attosecond pulse with a molecular system suddenly removes electrons, which can lead to significant changes in the chemical bonding and hence to rearrangements of the residual molecular cation. The timescales of the electronic and nuclear dynamics are usually very different, thus supporting separate treatment. However, when light nuclei are involved, as in most organic and biological molecules containing atomic hydrogen, the correlation between electronic and nuclear motion cannot be ignored. Using an advanced attosecond pump-probe spectroscopic method, we show that the coupling between electronic and nuclear motion in H-2 leaves a clear trace in the phase of the entangled electron-nuclear wave packet. This requires us to re-evaluate the physical meaning of the measured phase, which depends on the energy distribution between electrons and nuclei. The conclusions are supported by ab initio calculations that explicitly account for the coupling between electronic and nuclear dynamics.