We share common values and history with the Aussies, Kiwis and Brits and that is just about the end of it.
Our militaries are compatible. I suppose that if we decided to act together, we could form a formitable force in the world on our own without getting sucked up into the American coalition-du-jour and all of the negative consequences to those that follow them.
A CANZUK military force might save us some money. Shared resources.
But again, given our geographical distance, the advantage would still be limited, but still more good than harm.