Our world is constantly evolving and so are the societal issues and needs of its people. It's so easy to point to the bible and recite what it says about homosexuality, but it also says a lot about fornication, lying and backbiting. We are not supposed to judge each other. Compassion is what we are supposed to promote in each other's lives, and if two people love each other and want to be together then I think it should be encouraged.
If civil unions among bishops are in line with a church’s religious and or spiritual beliefs they should permit them to take place. However, if a church’s religious beliefs are not in accordance with such unions, to say they must permit them is just as unfair as saying those who agree or rather belief in such unions should not permit them. When it comes to religious or spiritual beliefs it is truly a matter of opinion.
In 1 Timothy 3, when Paul is instructing Timothy on appointing bishops in the churches, he tells Timothy, "An overseer [i.e. a bishop], then, must be above reproach, the husband of one wife, temperate, prudent, respectable, hospitable, able to teach, not addicted to wine or pugnacious, but gentle, peaceable, free from the love of money. He must be one who manages his own household well, keeping his children under control with all dignity (but if a man does not know how to manage his own household, how will he take care of the church of God?, and not a new convert, so that he will not become conceited and fall into the condemnation incurred by the devil. And he must have a good reputation with those outside the church, so that he will not fall into reproach and the snare of the devil." A person who was in a "civil partnership" would not qualify to be a Bishop.
Presuming "civil partnerships" refers solely to homosexual relations, the answer is obvious: No. The Bible outright states the following: "So God created man in his own image, in the image of God he created him; male and female he created them. And God blessed them. And God said to them, “Be fruitful and multiply and fill the earth and subdue it and have dominion over the fish of the sea and over the birds of the heavens and over every living thing that moves on the earth” (Genesis 1:27-28) and "But because of the temptation to sexual immorality, each man should have his own wife and each woman her own husband" (Corinthians 7:2). These two passages from the Bible outright state that it was God's intention to induce marriage between a man and a woman for the purpose of breeding. Therefore, under no circumstances should the Church sway from this. I'm no Bible thumper (heck, I've barely even read it), but I find it very hypocritical that the Church would even consider allowing this when its founding text outright discourages it. This has no bearing on my own personal beliefs about civil partnerships, but it reflects the Constitutionality of a law. If the Constitution contradicts a bill the President signs, the Supreme Court should throw it out. Likewise, if the Bible contradicts a church policy, the Church should throw it out. They need to practice what they preach, else they won't be taken seriously by anyone in ten years.