Steal part of the data field to hold a header extension with new 64-bit addresses (and other stuff if required)
Define a "magic" IPv4 address which indicates that the "real" extended IPvX addresses are at the start (or end) of the data field.
A true dual-stack IPv4/IPvX device would read the address in the header, and either route the packet as IPv4 from an old device or understand the "magic" address and retrieve the real addresses for correct IPvX routing.
In others words a (presumably transitional but not necessarily) gateway between OLD protocol, and NEW protocol, not a 'compatible protocol'.
Well, we now have the NEW protocol in IPv6 (though "new" is only relatively speaking) - nothing stopping you or anyone else writing the gateway you describe - at least, nothing that any "different IPvX" could solve.
The local stack would need to be altered... unless you just ran the conversion on your edge router... and then.... you've just designed a new form of IPv6 over IPv4 tunnel!