AHC: Southern Cone becomes Spanish "dominions"

The Southern Cone countries of Chile, Argentina and Uruguay are different from the rest of the former Spanish Empire in that they are populated primarily by whites.

How could these countries today have a similar relationship with Spain, as Australia, New Zealand and Canada have with the UK? Could they have become self-governing but retaining the Spanish monarch as head of state?

Of course, even if they do have this status originally, perhaps they'd still break from Spain completely if Spain becomes a republic as it did (briefly) during the 1930s...
 
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