I've heard from many people working in tourism or similar industries that, white Australians can be recognized as Australian solely by their facial features. Being Australian myself I've heard variants of the claim quite often and lean towards the claim being at least, a possibility.
While it's true that people tend to see patterns when there are none or incorrectly attribute, why they correctly recognize something. I don't necessarily think that is the case here. Australians don't tend to dress in a distinguishing way when overseas, and people have recognized them before hearing them speak. Given the extent to which I hear this claim made, I can't completely dismiss it.
Examples of claims
This article from an Australian newspaper in 1943 interviews an artist who when asked if Australians have a racial type, says he thinks that it was generally possible to distinguish an Australian [from other Caucasians] due to differences in the nose and differences in the jaw.
Also of note is the art project to show the average face of Sydney, of which the following quote is relevant:
In many cases, however, the likeness is so strong that it's possible to guess the nationality just by taking a cursory glance at the photo.
Reasons for thinking it may be possible
I'm fairly ignorant when it comes to biology, although there are a few reasons I think this might be possible. The (relatively) small gene pool from when Australia was colonized could have led to features from a small group being inherited in a large population.
There seems to be some evidence that environment can play a large role in influencing genetics more so than ethnicity. I can't find the paper investigating that, although will update this question if I am able to.
It seems possible to me that facial features belonging to the original colonists could have spread through subsequent populations, becoming a defining characteristic for some Australians.
I don't think most immigrants are relevant to this question as they will not have inherited distinguishing features (if they exist), nor will they or their parents/grandparents have been able to be influenced by the environment - if that is even a relevant factor.
Do a significant number of white Australians, excluding 1st or 2nd generation immigrants, have distinct facial features unique or generally only found in Australia?
Is it reasonable that some white Australians could be recognized as Australian, going only by facial features?