It doesn't seem to me that ACT and CT are incompatible. For argument's sake, let's assume that the two treatments can be merged in a meaningful way. If they can be merged, how would we go about basically integrating the different techniques (we'll save integrating the philosophies for a different day)? Well, some may argue that placing an emphasis on honing the ability to catch thoughts with the use of mindfulness techniques must precede the teaching of disputing irrational thoughts. At first glance this seems logical perhaps because it is assumed that one must be mindfully aware of a thought before being able to dispute it. I'm not so sure.
Through cognitive restructuring, if patients are first made aware of the fact that their thoughts are irrational, they may have a better framework in which to do ACT work. After cognitive restructuring, they have not only identified which thoughts are dysfunctional or irrational, but have been lead through a reasoning process of why these thoughts are inaccurate. Now with a deeper understanding of why their thoughts are irrational and a clinically significant lesser degree of belief in the verity of the content of these thoughts, they could be in a better position to recognize which cognitions are best to defuse from. Speaking practically from an ACT perspective, using the chessboard metaphor as an example, patients may better be able to identify the "black" pieces from which to defuse, while still being able to maintain self-as-context. After cognitive restructuring and focusing on self-as-context, it seems to me that people have a greater fighting chance of being able to accomplish the very cerebrally taxing feat of letting thoughts float by. In short, by first providing cognitive restructuring and framing the self as a container of both rational and irrational thoughts, increased understanding, use, and efficacy of mindfulness techniques may follow.
Of course I recognize that "irrational" thoughts are not the only type of cognitions associated with negative or dysregulated affect. There are a range of painful memories, facts, images, and maybe even manic tendencies with which disputation may not be particularly effective, and whereby defusion and mindfulness may be more appropriate; however, systematically disputing the cognitions associated with negative affect that are irrational before getting into ACT work may still generally be helpful.