As expected Vodafone email relaying is DENIED on a TPG platform. A few well accepted reasons for that.
The "mail.tpg.com.au" email server receiving SMTP traffic WILL ONLY RELAY TPG customers .....with a @TPG.com.au TLD.
HOWEVER if A Vodafone product is a PART of the TPG OFFERING (ie Mobile Broadband)
...That restriction SHOULD THEN NOT APPLY!!! ...BUT that restriction does apply.
One evidence is that a email@example.com email acct Also cannot AUTHENTICATE on Port 22 , Port 587, or Port 465 FROM a Mobile broadband connection to the mail.tpg.com.au server.
AFAIK A disclaimer to the fact that the two products do not behave as expected is not made by TPG sales advertising.
AGAIN: An email acct (firstname.lastname@example.org) can (Email) relay outside of the TPG mail server to another email server. Where as a (email@example.com) ie Non email acct CAN NOT.
Again ....if the (firstname.lastname@example.org) wants to relay thru TPG from the TPG product it should able to .......BUT not the case for the Mobile broadband product ..(which happens to be on the Vodafone network).