Historically speaking It would seem that different cultures handled Cognitive Dissonance differently.
The Rif and the Rambam - following the Gaonic Codifications - preferred a separate tome - thereby weaning the public away from the text of the Talmud towards works of Halachah. By this separation - if not outright divorce - Peace of Mind is attained. IOW the conflict is not camouflaged, it's sent away to a different venue entirely.
Both Rashi and Tosafot OTOH stayed with the Talmudic text and chose to comment upon it instead of to avoid this dissonance. Yet even with this common ground, they diverged for the most part.
Rashi - for the most part - makes highly subtle deviations from the text - so subtle only a sensitive scholar would notice.
Illustration: The Talmud in Shabbat 51b says that one may not crush snow or ice with one's hands - rather to place it in a cup. NB: This suggests a passive process is OK. Rashi [s.v. Aval notein letoch hakkoss] adds a subtle caveat - [only] when that cup contains wine [or perhaps another liquid]. Thus the issue morphs from passivity regarding melting ice - to issues of nolad by the subtle insertion of a word or 2.
Tosafot OTOH is rarely subtle. Tosafot's interpretations often SEEM or APPEAR at odds with the texts. This is unsettling for many. Fundamentalists chafe at Tosafot, while analytical or dialectical types celebrate it. Tosafot refuses to give up on the text - and also refuses to give up on "his" vision of how the Halachah should be.- no matter how disparate the two are. Tosafot doesn't avoid or side-step Cognitive Dissonance - rather "he" celebrates it!
We will explore this from another angle re: L'asuqei shmaat'ta aliba dehilcheta