We study electronic phase competition in a system of three coupled spinless Luttinger liquids using abelian bosonization, together with a perturbative renormalization group (RG) analysis. The scaling procedure generates off-diagonal contributions to the phase stiffness matrix, which require both rescaling as well as large rotations of the fields. These rotations, generally non-abelian in nature, are important for correctly obtaining the dominant electronic orders and critical behavior in different parameter regimes. They generate a coupling between different interaction channels even at the tree-level order in the coupling constant scaling equations. We study competing phases in this system, taking into account the aforementioned rotations, and determine its critical behavior in a variety of interaction parameter regimes where perturbative RG is possible. The phase boundaries are found to be of the Berezinskii-Kosterlitz-Thouless (BKT) type, and we specify the parameter regimes where valley-symmetry breaking, chiral orders, and restoration of $C_{3}$ symmetry may be observed. We discuss experimental systems where our approach and findings may be relevant.
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