The major offices of our government are either appointed by the Governor-General on the Prime Minister’s recommendation (which is binding in effect) or the House of Representatives. A quick look at the legislation shows an interesting array of methods, mainly between appointments on the Prime Minister’s or House of Representatives’ recommendations:

Chief Justice - “appointed on the recommendation of the Prime Minister.”
Chief Ombudsmen - “appointed by the Governor-General on the recommendation of the House of Representatives.”
Chief of Defence Force “The Governor-General in Council may from time to time appoint an officer of the Armed Forces to be the Chief of Defence Force.”
Auditor-General - “appointed by the Governor-General on the recommendation of the House of Representatives.”
Electoral Commission - “The Governor-General, on the recommendation of the House of Representatives, must appoint 3 members of the Electoral Commission as follows:
(a) 1 member as the Chief Electoral Officer; and
(b) 1 member as the chairperson; and
(c) 1 member as the deputy chairperson.
State Services Commissioner - “appointed by the Governor-General in Council on the recommendation of the Prime Minister.”

It seems odd then that the Governor-General’s appointment is undefined. At the moment, it’s essentially the formula for the Chief Justice - on the recommendation of the Prime Minister.