With "mains" and "240V", I assume you are in the UK?
Plugs do not have to be earthed as long as there is "double insulation" throughout, that is two insulators between the customer and anything live ("hot" as it is called in America). That can be two layers of PVC on normal mains flex, or the plastic equipment case and an air gap inside (in other words it is illegal to have anything live actually touching the plastic case.
I am a little puzzled too by the Earth-phase discrepancy as well. How close to each other are they? The reason I ask is that it sounds to me like they may be in a 3-phase building and each socket is actually on a different phase of the supply.
You could also have a ground loop forming by connecting the two Earths like that.
It does seem like that second Earth is actually getting the full 240V mains from the first socket, are you absolutely sure it was connected to the first socket's Earth and not live by accident? If the sockets are together and the building have only a single-phase supply, then I would say the floating "Earth" on that second socket is actually getting 240V mains on it from somewhere, like leakage current from some other equipment on that circuit. It could be very low current, but at 240V to give such a misleading result.
This is one of those occasions when a neon screwdriver used by hobbiests as a quick test to see if a circuit is live or not really comes in handy. If you do not have one of them, do you have it's modern equivalent of a "volt sniffer" or something like that? If you do, test that second Earth for mains voltage when connected to Earth one and when it is not. That should help you narrow it down a bit.
Good luck!