Generally, a webbing will be used that gets stapled to the stakes. Traditionally we would use Whites 50mm x 33m Jute Webbing Plant Tie but as an organic fibre, it will disintegrate after several years. You might wish to consider using Grunt 50mm x 10m Poly Webbing Black which I suspect is similar to what has been used there already. How many years did you get from the current webbing? I ask because staking is normally only necessary until the tree is fully established and it appears to be well on its way to doing so.
Let me mention @Adam_W to see what his go-to material is for tieing trees to stakes.
Please let me know if you need further information or had any questions.