= Solution
The velocity coupling changes the momenta and the undifferentiated field terms. Variation with respect to $\phi_2$ gives the equation for $\phi_1$:
$$
\frac{\partial\mathcal L}{\partial\phi_2}=-\frac{\phi_{1,t}}{2\tau},\qquad\frac{\partial\mathcal L}{\partial\phi_{2,t}}=\phi_{1,t}+\frac{\phi_1}{2\tau},
$$
so the two <Euler-Lagrange field equations> are
$$
\boxed{\phi_{1,tt}-c^2\phi_{1,xx}+\frac1\tau\phi_{1,t}=0,\qquad\phi_{2,tt}-c^2\phi_{2,xx}-\frac1\tau\phi_{2,t}=0.}
$$
They describe <damped and antidamped fields from an antisymmetric velocity coupling>. With the convention $\phi_j=A_je^{i(k_jx-\omega_jt)}$, their characteristic equations are
$$
\omega_1^2+\frac i\tau\omega_1-c^2k_1^2=0,\qquad\omega_2^2-\frac i\tau\omega_2-c^2k_2^2=0.
$$
Thus the <dispersion relations> and modes are
$$
\boxed{\omega_{1,\pm}=-\frac i{2\tau}\pm\sqrt{c^2k_1^2-\frac1{4\tau^2}},\qquad\phi_{1,\pm}=A_{1,\pm}e^{i(k_1x-\omega_{1,\pm}t)},}
$$
$$
\boxed{\omega_{2,\pm}=+\frac i{2\tau}\pm\sqrt{c^2k_2^2-\frac1{4\tau^2}},\qquad\phi_{2,\pm}=A_{2,\pm}e^{i(k_2x-\omega_{2,\pm}t)}.}
$$
For each spatial <Fourier mode>, the general solution is the <linear combination> of its two temporal branches; spatial superposition gives the general field. Take <real parts> or suitable <complex conjugate> pairs to obtain the required real fields. At the repeated-root threshold, the independent second temporal solution contains an extra factor $t$, as detailed next.
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