The Commonwealth armies, remembering their WW1 & Boer War experiences, had been issued very clear orders from their own governments, communicated to the British, that they were not to fritter their strength away in penny-packets or in engagements in which they were not properly supported. Freyberg had definitely been granted written permission to extract the 2nd New Zealand Division, in effect the entire New Zealand field army, from combat if he deemed it necessary. On at least one occasion he did so, following the failure, yet again, of British armour to support his division despite being assigned to do so. The South Africans had similar orders and were also not to be used outside of Africa. The Australians, Canadians, New Foundlanders, and Indians also had conditions attached to the use of their armies in British Imperial service. Australia even withdrew its divisions from North Africa to defend against the Japanese in the Pacific. New Zealand very nearly followed that example, only remaining in the Mediterranean Theatre due to a combination of pressure and promises from both the British & Americans.
So, despite what their allies and enemies may have thought, the British Commonwealth & Imperial armies were not monolithic, but were in both fact & practice coalition armies that required negotiations at the highest levels regarding their deployment in the field.