It's called election. In other words, why not devise a system to elect bishops? Let the people of a diocese choose their bishop(s). Now, this would require some process for nominations, probably presentations by the candidates either in a public forum or at least in writing, and formal voting. Eligibility for voting ought to include both women and men in good standing in a given diocese ... probably 18 years of age or older, as in our secular election system. Or, if one wanted to dilute the democratic process a bit, election might be done by lay delegates who are elected by parishioners.
In any event, this would revive the system for bishop selection that predominated in the earliest centuries of the church. An early history of this practice by Juicio Brennan says that, "In these early centuries, the nominations and elections of bishops were done solely by a popular vote of all the faithful. St. Cyprian believed elections prevented unworthy persons from becoming bishops." He also notes that "by the middle of the third century ... evidence shows that women were beginning to be excluded from the voting." Obviously, that would be unacceptable today.
Religious communities of women (and most men's communities as well) elect their leaders. So it's way past time to think about democratic processes in our church at large.
Sr Maureen Fiedler; National Catholic Reporter; 15 September 2016