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Kim Zolciak & Kroy Biermann

Kim Zolciak and Kroy Biermann's divorce has telltale signs it's already super nasty ... with a war brewing over the estranged couple's children and broken finances.

We've now obtained Kroy's petition for divorce, and he wants sole legal and physical custody of the pair's 4 minor children. That in itself is an aggressive move, essentially cutting his estranged wife out of the child-rearing process.

Kim's legal docs also reveal things are not kumbaya. She's asking for sole physical custody with visitation for Kroy.