Thompson's group $F$ is totally ordered.  See [here][1] for a description of all possible bi-orderings.  Indeed, all diagram groups are totally orderable (see [here][2]).  The pure braided Thompson group $BF$ also bi-orderable (see [here][3]).


  [1]: https://arxiv.org/pdf/0808.1688.pdf
  [2]: https://arxiv.org/pdf/math/0305153v1.pdf
  [3]: https://mat-web.upc.edu/people/pep.burillo/Papers/biorder.pdf