No, I don't think it's possible for any permissioned blockchain platform today to share a blockchain with a different platform – the industry is a long way from this level of standardization. And I'm not sure it should be aiming for it any time soon, since each blockchain platform has different capabilities, a different consensus model, different cryptography and so on – these are things which effectively prevent standardization. It's a bit like want a Git client to be able to connect to a Subversion repository.
