across the ditch

By analogy with across the pond, and the fact that the Tasman Sea is much smaller than the Atlantic Ocean.

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across the ditch

1. (informal, Australia, New Zealand) On or to the other side of the Tasman Sea.
   • 2016, Peter Mares, Not Quite Australian: How Temporary Migration Is Changing the Nation, →ISBN:

     The rate of arrivals fluctuates with the relative economic fortunes of each country, but since the late 1960s, faster growth, higher wages, greater job opportunities and a stronger dollar have been an economic magnet drawing New Zealanders across the ditch.