Intriguing phases may emerge when two-dimensional systems are coupled in a bilayer configuration. In particular, a Berezinskii-Kosterlitz-Thouless (BKT) paired superfluid phase was predicted and claimed to be numerically observed in a coupled $XY$ model with ferromagnetic interlayer interactions, as reported in Phys. Rev. Lett. 123, 100601 (2019). However, both our Monte Carlo simulations and analytical analysis show that this model does not exhibit a BKT paired phase. We then propose a new model incorporating paired-phase gradient interlayer interactions to realize the BKT paired phase. Moreover, we observe that the anomalous magnetic dimension varies along the phase transition line between the disordered normal phase and the BKT paired phase. This finding requires an understanding beyond the conventional phase transition theory.

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• BKT transition
• Classical spin models
• Monte Carlo methods
• XY model