Answer: A self clinging vine such as English ivy (Hedera helix) would be able to climb rock without any extra support. However, a more cold hardy plant to consider is Virginia Creeper, (Parthenocissus quinquefolia). This vine is deciduous but it turns a beautiful red in the fall. It would also be able to cling to the rock without additional support.
Since the cliff is eroding badly, I would strongly suggest you consult with an engineer or other professional grading expert to make sure that it is safe as it is, and possibly to suggest an alternative direction/control pattern for the runoff.
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