I am doing my Ph.D. in Australia and so you have two examiners. I sent my thesis for examination last year and got it back with one supervisor saying it was a pass but one said R+R and the adjudicator agreed on the R+R. This was based on some statistics that the examiner didn't quite understand. My supervisor convinced me that I was right and that I should make it slightly clearer and send it out again. I got it back the second time (from the same supervisors) with one saying pass and the other failing it. Now I have to send it out to an adjudicator again and I am worried they will side with the examiner that failed me again. Any advice?

It isn't official yet. There is nothing you can do, just relax.

You are worried they will fail you but that hasn't happened yet. You can't start an appeal until you get the results. It sounds like you like your supervisor but do you have any grounds to appeal? As it is hard to appeal the actual decision unless their is an underlying issue or clear conflict. There are a couple of people on the forum who have done the appeals process and you might their attention with a well written title. If you do choose to appeal you should talk with someone in your department for advice.

For all you know the adjudicator could pass you (with corrections) and the worry will be for nothing. What is it in particular do you want advice on?