Unfortunately,eucalyptus are shallow rooted and wind rock can expose the rootball very easy-mine a few years ago did the same.
You will need to stake it to keep the tree firm in the ground,but keep the stake very low down.This way,the top of the tree keeps moving in the wind and it encourages it to put down more roots to anchor itself in the soil.Lean,the stake at 45 degrees into the wind,hammer into soil-then attach the strap to tree.Cut any excess timber from post afterwards to neaten it up and also stops you walking into it!
Keep an eye on the strap though and loosen it off every so often so it dont cut into the trunk when it grows.In a few years,you may be able to remove the stake when its firmly anchored.
ps.Mine is 5 years old and taller than my house!