Are Charles and Camilla married?

Since members of the royal family are exempt from the Marriages Act (1949) it would seem that their marriage was valid only if, (since they were married in England) they were married in a ceremony of the Church of England (which, of course, they were not -- though I don't know whether the reasons why not were to do with Charles or to do with Camilla). The present government, unlike its predecessor, thinks, however, that the Human-Rights Act 1998 (which, ironically, Charles himself, in a letter to the previous Lord Chancellor, denounced as “a threat to sane, civilised and ordered existence”) mandates a liberal interpretation of the Marriages Act such that Charles's human rights should not be infringed by his being prevented from contracting a civil marriage.

There is another question, of course, as to whether Charles could have married her had he been King and thereby Supreme Governor of the Church of England (since it was apparently for this reason that the Queen felt that she couldn't be present at the ceremony): Edward VIII famously had to abdicate in order to marry Mrs Simpson.