Your description 'She said this morning that she isn't allowed to play with other people because G gets upset' rang bells with me.
The problem I mentioned in my previous post, concerning my older daughter (so no twin involved in that instance) was very similar.
A girl was isolating her from everyone else and monopolizing her and not allowing her, sometimes physically, from having other friends. Dd used to say the girl would 'get cross with me' when she tried to play with others. The girl was clearly clinging to my dd because she had not made other friends. I felt sorry about that but my dd was suffering because of it too.
My dd became withdrawn and unhappy. The teacher said she would spilt them in class, but of course it's the playground where it all happens.
They informed the staff supervising at playtimes and my dd was told that if she was unhappy to go directly to one of them and they would deal with it and diffuse the situation. This strategy seemed to be working a bit but just as I felt I might need to up the ante, the girl happened to leave the school anyway, so I can't give a definitive start to end solution to this - although to say dd was relieved is an understatement!
I think the fact that other teachers and dinner-ladies knowing about it gave my dd adults to turn to in the playground who wouldn't dismiss it as a squabble or telling tales.
My dd was always at pains to still play with her friend because she liked her and wanted to be a friend, but the minute her free will was being compromised she knew she could turn to an adult. If irrc the idea was that the girl could play with my dd alongside others and therefore would begin to have the confidence to make other friends of her own.