So the Australian Army lacks royal sanction?
just kidding, bless the great ADF!!
Australia adopted the same tradition in naming our armed forces (so did Canada and New Zealand) - Air Force and Navy are Royal - Army isn't.
Individual Army Regiments can bear Royal titles (so we have the Royal Australian Regiment, the Royal Australian Artillery, and the Prince of Wales's Light Horse for example) but never the Army as a whole.
Yeah it comes from the British Army (also without Royal sanction) being formed as Monck’s Regiment of Foot in 1650, under Cromwell. Some antecedents cannot be forgiven