Many claims are made about the Covenant and what it will do and not do, most of which are hypothetical claims because, let's remember, the Covenant is a proposal, not a reality. But these claims fuel the fire of fears so they are particularly effective since emotions are strong arguments for agreeing or not agreeing to a course of action. One might even venture a little further and consider whether arguments for the Covenant are essentially theological (the unity of the churches in Communion is agreeable to Christ) and arguments against the Covenant are essentially emotional (it will be the cause of great upset, wailing and gnashing of teeth).
One of the most damaging claims about what the Covenant will do is the claim that it will usher in an Anglican papacy, curia and magisterium. This is a particularly damaging claim because it plays on the visceral fear at the heart of Anglicanism that we might one day again be subservient to Rome. In one of those quirks of life, fear of papacy is a uniting force for both Anglican liberals and conservatives, as well as all those in between. Not that the Covenant will make us subservient to Rome, but even the possibility of an Anglo-Papacy strikes terror in the Anglican heart!
But the Covenant will not usher in any kind of papacy, at least not any kind not already familiar to us. The Covenant is simply about the Communion enjoying what every member church already enjoys, a written constitution and code of canons constraining licensed clergy and officers of the church to live by that constitution. Are member churches run by popes? Do they have curia and magisteria? No! Nor will the Communion if it embraces the Covenant.
How about we stop lying about the Covenant by desisting from playing on fears of papacy?
Incidentally there is one slight amount of popery already in the Anglican system, but it will not change one whit with the Covenant in place: the Archbishop of Canterbury has discretion about to invite or not to invite to the Lambeth Conference.