Paul's Defeat at Antioch
Stephen Carlson asks a question we've heard before, Did Paul Win at Antioch?, and lists the pros and cons for "yes", "no", and "maybe so". I'm a confident naysayer: Paul lost at Antioch, just as he later did in Galatia. That he would have played the trump card overrides the counter-arguments. Philip Esler also notes that the biblical phrase, "I opposed him to his face" usually occurs in the context of unsuccessful resistance (Galatians, pp 135, 140).
The reason Paul would have brought up his defeat, as Stephen asks, is (1) to illustrate that he (unlike Cephas) is consistent with his preaching of the gospel (addressing in part the charge of Gal 5:11); and (2) the Galatians probably knew of it anyway (gossip had a long arm in the ancient Mediterranean), and he simply couldn't ignore it in the context of the current crisis which centered on the same question: circumcision (Gal 2:12) (not dietary laws). Paul is trying to save face as best he can, but it's hard to obscure the fact that he lost after James and Peter broke the agreement.
Paul's failures in Antioch and Galatia, and increasingly bad reputation, were responsible for his rhetorical shifts in Romans, where he found ways to respect Judean identity even while insisting that the law and covenant were completely obsolete.