I am new to this Forum business so let me know if I have given enough information.
The "thing" appears to be growing near the base, around the stem of a climber (a clematis I think, don't know the name but it has little bunches of deep red/chocolate flowers) in my garden on the Surrey/Hampshire borders. It is well protected and south facing. The ball are about 3cm across and the larger open one on the right is about 6cm long by 3cm.
I hope that helps but as the concensus is that it is a fruit I will post it there as well.