I'm not an advanced programmer so I don't know many of the technical details, but it seems something like this could be possible.
It isn't impossible, it's just that it would require very advanced programming expertise and some changes to the Bitcoin protocol to be able to explicitly track sub-variants of Bitcoin again double-networking (currently Bitcoin guards agains double-spends, it would need to safely guard against transferring same bitcoins to more than one sub-network).
If you want you can take a look at my old proposal from my signature:
https://bitcointalk.org/index.php?topic=54382.0But I have to warn you that advanced programming knowledge is required to understand it.
Currently there's absolutely no consensus on how to change the Bitcoin's network protocol.
The one thing I don't like is the title of this thread: it should be "Competing and cooperating Bitcoin networks". They are competing in serving of customers but simultaneously cooperating against fraud and counterfeiting.
Edit: Without that last caveat the purely "Competing Bitcoin Newtorks" would be just Alternative Cryptocoin Networks".