The via media theory states that Anglicanism is neither purely Protestant nor purely Catholic (in the Roman sense) but some sort of "middle way" between the two systems. Usually the term is used to state that the Anglican Churches have maintained the best of Protestantism and the best of Romanism. The rhetoric also includes a moderateness whereby Anglicans rejected the extremes of these other systems and instead opted for this middle way. For an example, here's the Wiki entry on Anglicanism:
Now, I'm going to show a few quotes from the Articles of Religion and the reader can decide whether or not the Anglican Reformers wanted anything to do with Romanism.
Anglicanism, in its structures, theology and forms of worship, is commonly understood as a distinct Christian tradition representing a middle ground between what are perceived to be the extremes of the claims of 16th century Roman Catholicism and the Lutheran and Reformed varieties of Protestantism of that era. As such, it is often referred to as being a via media (or "middle way") between these traditions.
The Bishop of Rome hath no jurisdiction in this Realm of England.Mind you this is not an exhaustive list, only the explicit mentions of the Roman Church in the Articles of Religion. This strong denunciation of Roman doctrine continued after the Reformation. The Caroline Divines were firmly against Roman innovation in doctrinal matters. Even into the 18th and 19th centuries, Anglican consensus was strongly against Romanism. The real change begins to happen in the Oxford Movement (but also just before with Alexander Knox -- who espouses justification by infusion). The Oxford Movement began as a movement to take seriously the catholic nature of the Church of England but ended up destroying that Church in the short-term and long-term.
That's right the via media theory begins with John Henry Newman. The iconic Tractarian and convert to Roman Catholicism. Newman invented this phrase to accommodate his theological views in the Church of England, when the formularies clearly rejected them. He later recanted this idea, stating that it was a dream, and converted to Rome. One wonders why we are continuing to use a theological idea that was created in the 19th century by a man whose loyalty to Anglicanism was questionable at best (during his time in the Church of England) and who later disproved the idea and converted to Rome.
Now, because I am saying that the via media is garbage doesn't mean that I am saying that the Anglican Churches are not Catholic. Let me briefly explain the real meaning of that term. The Church of England and her daughter Churches claim to be Catholic. That is not due to proximity to the doctrines of Rome or Constantinople but because, at the Reformation, the doctrines of the early Church were recovered and the innovations of the Middle Ages were removed from the Church. The Church of England is a Reformed Church and due to that is a Catholic Church. The two terms are synonyms, to be Reformed is to be Catholic, by nature, and vice versa.