It's a strange bug but don't worry, as soon as the person you're EB'd to will log in you will see their name again on the ring.

I had that happen myself where it said "Obtaining Signature" so i told my partner to log in, their name showed up again on the ring once they did but strangely enough it vanished again like a day or two later. It seem to happen when your partner doesn't log in for 24h or 48h. In my case my EB ring still says "Obtaining signature" but thats because they didnt log in in a while.