Many physicists use 'reality cannot have a true paradox, only apparent ones in our theory' as a very common maximum and guide to theory checking. As with all universal rules, Paul Feyerabend disagrees that this is useful rule across science, and suggests that there are facts that are best described by inconsistent theories.
Unfortunately he only gives one such example: optical illusions where a static object appears to be moving. He asserts that "[t]he only phenomenologically adequate description is 'it moves, in space, but it does not change place'".
In a footnote (ft 92, ch 16) Feyerabend tries to address the obvious criticism (raised to him by A.J Ayer and G.E.L. Owen) that the above statement deals with appearances by saying that putting "it appears to move" in front of the statement just pushes the question to the level of appearences but doesn't resolve it.
However, it seems to me that Feyerabend is completely missing the point, because he is assuming that we are saying (with glorious use of brackets added by me) that:
[A] "(it appears that it moves in space but does not change place)"
but any person actually making that statement actually means:
[B] "(it appears that it moves in space) but does not change place".
I would agree with Feyerabend that the statement [A] doesn't achieve anything except introduce an extra judgement statement, but statement [B] (which is what I think most people mean) separates two domains of 'appearance' and 'reality' and thus is not contradictory.
Feyerabend spends a lot of time talking about appearance and reality elsewhere, so I am worried that maybe I am missing some subtlety that makes statement [B] contradictory. Am I missing some subtlety in Feyerabend's critique of self-consistency?
Alternatively, if this particular argument is so weak then it seems like for such a bold claim he would support it more elsewhere. Are there other articles where Feyerabend (or subsequent authors) more closely defends his critique of self-consistency? Alternatively, are there more convincing examples of the failure of self-consistency?
Sources and further information:
 In his typically colourful language he writes (on pg. 245-6 of 2010 reprint of Against Method):
the idea that things are well defined and that we do not live in a paradoxical world leads to the standard that our knowledge must be self-consistent. Theories that contain contradictions cannot be part of science. This apparently quite fundamental standard which many philosophers accept as unhesitatingly as Catholics once accepted the dogma of the immaculate conception of the Virgin loses its authority the moment we find that there are facts whose only adequate description is inconsistent and that inconsistent theories may be fruitful and easy to handle while the attempt to make them conform to the demands of consistency creates useless and unwieldy monsters.
 His actual example is on pg. 202:
fixate on a moving pattern that has come to a standstill, and you will see it move in the opposite direction, but without changing position.
He would not approve of my use of the word 'optical-illusion' in the description, so I don't mean to smuggle in the naturalist interpretation, I am just giving the easiest way to remind you of the phenomena. I think the Rotating Snakes would be a fine and easier to see example of the sort he meant.