As was suggested in a previous post, I used a square cut scraper to try and remove the tearout.
Having looked at the wood in more detail and some other pieces that I have cut, I am unsure that it is even fair to call it tearout. The wood is so 'rotten' that I really do not think that I will achieve very much in the way of reducing tearout.
I am in the middle of turning another piece right now and feel that the fibres are extremely loose in some areas, particularly the end grain. Super Glue or wood hardener as suggested by John BHT is quite possibly the way forward or some way of stabilising the wood.