How to measure Projective Symmetry Group in spin liuquid?

Quasiparticles in spin liquid will no longer be the representation of symmetry group. So when group elements act on quasiparticles, there will be some phase factor. For example, in $\pi$ flux state, $T_xT_y=-T_yT_x$, which means if we move the quasiparticle around a lattice, and there will be a minus sign. However, because the quasiparticle are always emerging in pairs, when we move one quasiparticle, it cannot be avoided that we will also move another. In this case, the phase factor (minus sign) will be cancelled out. If we wants to move only one quasiparticle, the symmetry operator must be a local operator. But what does it mean for local operator of the global symmetry? How can we measure the PSG in general cases?

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One can partially and indirectly measure PSG by measure the spectrum function of spin correlation (which is a two spinon spectral function). In π flux state, $T_xT_y=−T_yT_x$, the lower edge of the spectral function is repeated in 1/4 of BZ. (See Fig. 11 of arXiv:cond-mat/0107071, for example)