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\begin{align}  \Theta_I &=& \phi_I - \phi_{N+I} \\  \Phi_I &=& \phi_I + \phi_{N+I} \\  \mathcal{L}_\text{nc} &=& \frac{1}{8\pi} \left[ 2  \partial_x \Theta_I \partial_t \Phi_I + ... + 4e A_t^{(I)} \partial_x \Phi_I + ...\right] \end{align}  Then ... and if (crucially) I find the Noether charge of the non-chiral action,  I get $$ P = \frac{1}{2\pi} \sum_I k_{F,I} \left(\partial_x \phi_I - \partial_x \phi_{N+I} \right) + \frac{1}{4\pi} \sum_I \left( (\partial_x \phi_I)^2 - (\partial_x \phi_{N+I})^2 \right) $$