If I send you 100 BTC's, from wallet abc to your efg, what split would occur naturally? If the network were split, potentially both could have the same spend... right? But they would be to the same wallet.
If you double-spend, then both recipients will get transactions with 0 confirmations. The network isn't sure yet which one came first, since the simple TCP network can't be counted on. Normally after one block the network will decide which one is valid. However, it is possible (though unlikely) for two blocks containing different versions of the transaction to be generated at the same time, in which case it would take another block to figure out which one is valid. (It's possible for this to happen a second time, but too unlikely to think about.)
So a detection mechanism could say, "This new block would replace a block that is 5 blocks deep in the main chain. Any transactions in that chain that conflict with transactions in the main chain are likely to be double-spends." It could not say (with reliable accuracy), "This new block is now the latest block in the main chain. However, one of its transactions conflicts with one that I received via the TCP network first, so the block's version must be a double-spend."