The legislation makes clear that an alternative male bishop would not be independent of his female diocesan bishop but that his authority would instead be "delegated" to him.
It adds that delegation should not be taken as divesting the diocesan bishop of any of their authority or functions.
The legislation states further that the Code of Practice will include guidance for diocesan bishops on the selection of male bishops and priests to ensure their theological convictions reflect those of the parishes that have requested oversight.
It reads: "Thus, the legislation now addresses the fact that for some parishes a male bishop or male priest is necessary but not sufficient.
"The House rejected more far-reaching amendments that would have changed the legal basis on which bishops would exercise authority when ministering to parishes unable to receive the ministry of female bishops."
The issue of women bishops has deeply divided the Church of England, although the draft legislation presented to the bishops was approved by 42 out of the 44 dioceses last year.
It is not only male traditionalists who are opposed to the consecration of women bishops.
A petition was presented at the start of the meeting in York yesterday that was signed by more than 2,200 Anglican women who oppose women bishops.
The petition was organised by General Synod member Susie Leafe who said: “Not all the women in the Church of England think having women bishops is a great idea - our petition proves that, and we ask our bishops to recognise that and make proper provision for us.
"We believe that God created men and women equal but different, and that those differences are seen in the God-given roles that men and women have within the family and within God’s household, the church.
“This is not an outdated view held by a few ‘diehard traditionalists’. Our survey of those who signed the petition shows that they come from churches that are growing, youthful and very female friendly.”
Some traditionalists have already left the Church of England over the consecration of women bishops and joined an Ordinariate set up by the Roman Catholic Church.
Supporters of women bishops argue that male oversight for traditionalist parishes is a significant compromise, but some traditionalists may still feel that the provision in the draft legislation does not go far enough.
As the legislation has been amended, the six Officers of the Synod, who include the Archbishops of Canterbury and York, will meet later in the week to determine whether the legislation now differs in substance to that approved by the dioceses last year.
If they approve the amendment made by the House of Bishops, the legislation will be returned to the General Synod for final approval when it meets in York in July.