The only clue we are given in 'A Good Man Goes to War' is that the Papal mainframe is female (in terms of some aspect of her personality, origin or honorific title presumably). We are also shown that she retains considerable clout within the militarised structure of the Clerics. Thus she is able to give Colonel Manton special permission to unveil the identity of the Hooded Monks. 

In terms of real world context this is a bit of a cheeky dig at the present day Catholic church. While other Christian denominations have made some moves towards introducing or considering female priests and bishops the Catholic church has remained conservatively opposed. Apparently in the future of Doctor Who this stance has altered to the extent that the Church allows its (real or ceremonial) leader to be both female and non-biological.