What did you use to get a version number?
David A. sent me RtlChipId.exe, some time ago, for a previous Realtek NIC problem. I am not sure what the licensing is for that, but I am sure he would send it to you, if you are subscribed to Arca Noae.
I had a lot of fun trying to get this nic working until I discovered r8169-1.0.0.wpi in the eCS betazone. If you have not tried that build it may be worth a go.
I tried that one, and the Arca Noae equivalent. No change. I have 3 other Realtek NICS that work well with the Realtek driver from Multimac-20150529.exe (from Arca Noae).
I will also note, that I have a couple of Intel NICs, that work very well with the new E1000B driver, from Arca Noae (based on BSD code). Hopefully, they can get the Realtek NIC driver updated, soon. Well worth the cost of admission.