It appears Anglicans claim apostolic succession. How do they trace their apostolic succession?
The same way that the Catholics do.
Anglicans consider themselves catholic and reformed. The Anglican church was the Catholic Church in England until the reformation when they stopped recognizing the authority of the Pope.
"The roots of the Church of England go back to the time of the Roman Empire when Christianity entered the Roman province of Britain. Through the influences of St Alban, St Illtud, St Ninian, St Patrick and, later, St Augustine, St Aidan and St Cuthbert, the Church of England developed, acknowledging the authority of the Pope until the Reformation in the 16th century." (An Ancient Church, Catholic and Reformed - www.churchofengland.org)
They trace their apostolic succession through the bishops who were members of the historical episcopate that remained in England after the schism.
Up until the time of the Reformation, the line of apostolic succession is the same between the Church of England and Roman Catholicism. The English Reformation initially had more in common with the historic schisms* than with other iterations of the Reformation, though through the influence of Thomas Cranmer, the first post-Reformation Archbishop of Canterbury, it quickly took on a Protestant character.
Since the Reformation, the exact nature of apostolic succession has been under dispute within the Anglican Communion, and therefore there is no particular line through which all Anglicans can "trace" succession. However, as with many doctrines, all Anglican churches affirm the doctrine while granting freedom to observe it in different ways.
For example, the Chicago-Lambeth Quadrilateral, an "affirmation of Anglican identity" adopted by an international conference of bishops in 1888, states one of its four points of identity as, "The historic episcopate, locally adapted." Philip Thomas examined this in a 1998 Anvil magazine article of the same name.
Thomas looked at four ways in which the episcopate has been "locally adapted" throughout the communion:
- the 'democratising' of the episcopate in the founding constitution of the American Episcopal Church - and the example that this provided for other autonomous Anglican Churches which emerged during the nineteenth century;
- the way in which the ethnic experiences of tribal eldership have been incorporated into the structures of (for instance) the church in Melanesia, and the Province of Aotearoa/New Zealand;
- the challenge which African traditional religions have posed for western Christian institutions, leading to (in John Pobee's estimation) a more 'charismatic' expectation of the episcopate;
- the influence of prevailing political realities on the way leadership patterns are developed, e.g. the 'federalism' of Australian Anglicanism; the 'Westminster' ethos of the Church of England; the reflection of and reaction to tribal factors in parts of East and Central Africa.
While the Wikipedia article mentions three views of the role of the historic episcopate, for our purposes it's probably sufficient to consider two strands of thought: the low church view, and the high church view.
The low church view, articulated by Thomas (ibid.), says, "The connectedness of the church to the apostolic community is guaranteed by being 'in Christ,' not by the possession of particular norms of ministry or doctrine alone. ... The forms are not the reality, yet the reality itself is not formless."
The high church, or Anglo-Catholic, view says on the other hand, as articulated by Edward Goulburn in The Holy Catholic Church, "There is, and can be no real and true Church apart from the one society which the apostles founded and which has been propagated only in the line of the episcopal succession."
You could thus say that the high church view is "the historic episcopate, locally adapted," whereas the low church view is "the historic episcopate, locally adapted." (Note the differing emphases.)
Even for Anglo-Catholics, I have been unable to find a resource which lists the "unbroken line" of succession. But given that the much larger church communion of Roman Catholicism has gaps (however few) in its own succession records, this should hardly be surprising.