\section*{Supplemental Material }



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In this Supplemental Material, we present additional details and calculations regarding: 1) derivation of the evolution operator for the left and right state vectors in the non-Hermitian case; 2) Fourier transformation of the anomalous Green's functions; 3) electron-electron interaction mediated by phonons from two bands with strong asymmetric interaction. 

\section{S1. Derivation of evolution operators for left and right state vectors in non-Hermitian case}
In this section, we derive the evolution operators for the left and right state vectors. As we consider the non-Hermitian case, it is not obvious how they look like.

The $\eta$-pseudo-Hermiticity is defined as \cite{mostafazadeh:jmp02}
\begin{eqnarray}
\label{eq:etaH}
H^\dagger=\eta H\eta^{-1},
\end{eqnarray}
where $\eta$ is a Hermitian and invertible linear operator. Then the $\eta$-product is defined as
\begin{eqnarray}
\langle\Phi|\Phi'\rangle_\eta=\langle\Phi|\eta|\Phi'\rangle.
\end{eqnarray}
The $\eta$-product can be derived from the definition of pseudo-Hermiticity and Schr\"odinger equation as follows. The Schr\"odinger equation for the right state vector is
\begin{eqnarray}
i\frac{\partial}{\partial t}|\Phi_R\rangle=H|\Phi_R\rangle.
\end{eqnarray}
If we take Hermitian conjugation of it and substitute the definition of $H^\dagger$ from Eq. (\ref{eq:etaH}), we obtain
\begin{eqnarray}
-i\frac{\partial}{\partial t}\langle\Phi_R|\eta=\langle\Phi_R|\eta H,
\end{eqnarray}
that gives
\begin{eqnarray}
\label{eq:SchroedPhiL}
-i\frac{\partial}{\partial t}\langle\Phi_L|=\langle\Phi_L|H.
\end{eqnarray}
Now let's derive the evolution operator in the interaction representation. For that, we represent the Hamiltonian in free and perturbation parts, $H=H_0+V$. Then, we use the Schr\"odinger equation following the standard procedure:
\begin{eqnarray}
i\frac{\partial}{\partial t}[e^{-iH_0t}|\Phi^I_R\rangle]=i[-iH_0e^{-iH_0t}|\Phi^I_R\rangle+e^{-iH_0t}\partial_t|\Phi^I_R\rangle ]=(H_0+V)|\Phi_R\rangle.
\end{eqnarray}
The terms with $H_0$ cancel each other and we obtain
\begin{eqnarray}
i\partial_t|\Phi^I_R\rangle =e^{iH_0t}Ve^{-iH_0t}e^{iH_0t}|\Phi_R\rangle=V^I|\Phi^I_R\rangle.
\end{eqnarray}
This is a standard expression of the Schr\"odinger equation in the interaction picture. We can do exactly the same with Eq. (\ref{eq:SchroedPhiL}) with the only change that $\langle\Phi_L^I|=\langle\Phi_L|e^{-iH_0t}$ and obtain
\begin{eqnarray}
-i\frac{\partial}{\partial t}\langle\Phi_L^I|=\langle\Phi_L|e^{-iH_0t}e^{iH_0t} Ve^{-iH_0t}=\langle\Phi_L^I|V^I.
\end{eqnarray}
the evolution operator for $\langle \Phi_L^I|$ is
\begin{eqnarray}
\mathcal{S}_L=\bar{T}e^{i\int_0^td t'V^I(t')},
\end{eqnarray}
and for $|\Phi_R^I\rangle$ is
\begin{eqnarray}
\mathcal{S}_R=Te^{-i\int_0^td t'V^I(t')}.
\end{eqnarray}

\section{S2. Fourier transformation of the anomalous Green's functions}
The poles of the Green's functions derived in Eq. (8) of the main text are $\omega_{1,2}=\pm\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}$. We perform the corresponding integration following the standard procedure and firstly add $\pm i\eta$ to the denominators of the Green's functions with $\eta$ being infinitesimally small, $\eta\rightarrow+0$. As the electron Green's function usually has $i\eta$ and the hole Green's function has $-i\eta$, we put $i\eta$ to the denominator of $\bar{F}$ and $-i\eta$ to the denominator of $F$. If we express denominators in terms of products and expand the square roots in $\eta$, we obtain
\begin{eqnarray}
	\bar{F}_{\uparrow,\downarrow}^{{\bf k},-{\bf k}}(\omega)&=&\frac{\Delta_{\bf k}}{[\omega-\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}+i\eta][\omega+\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}-i\eta]}, \\
	F_{\downarrow,\uparrow}^{-{\bf k},{\bf k}}(\omega)&=&\frac{-\Delta_{\bf k}}{[\omega-\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}-i\eta][\omega+\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}+i\eta]}.
\end{eqnarray}
This implies that $\bar{F}_{\uparrow,\downarrow}^{{\bf k},-{\bf k}}$ has a pole $\omega_2+i\eta$ and $F_{\downarrow,\uparrow}^{-{\bf k},{\bf k}}$ has a pole $\omega_1+i\eta$ in the upper half plane. Building a contour of half-circle shape with the radius $R\rightarrow\infty$ and employing residues (see e.g. Ref. \onlinecite{kornich:prr21}), we obtain for the anomalous Green's functions in the time domain
\begin{eqnarray}
\bar{F}_{\uparrow,\downarrow}^{{\bf k},-{\bf k}}(t)&=&-\frac{i\Delta_{\bf k}e^{-it\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}}}{2\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}},\\
F_{\downarrow,\uparrow}^{-{\bf k},{\bf k}}(t)&=&-\frac{i\Delta_{\bf k}e^{it\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}}}{2\sqrt{\varepsilon_k^2-\Delta_{\bf k}^2}}.
\end{eqnarray}
We can see that $\bar{F}_{\uparrow,\downarrow}^{{\bf k},-{\bf k}}(t)\neq (F_{\downarrow,\uparrow}^{-{\bf k},{\bf k}}(t))^\dagger$.




\section{S3. Derivation of Electron-electron interaction mediated by phonons from two interacting bands}
Let's consider electrons interacting with phonons. Then, the partition function is given by
\begin{eqnarray}
\mathcal{Z}=\int \mathcal{D}[\bar{\psi},\psi,\bar{a},a]\exp{[i(S_{ph}+S_e+S_{e-ph})]},
\end{eqnarray}
where $S_e$ is the action of electrons with $\psi$ and $\bar{\psi}$ being Grassmann fields of electrons and $S_{ph}$ is the action of phonons with $\bar{a}$ and $a$ being bosonic fields of phonons. We make a common assumption that $\langle a\rangle=\langle\bar{a}\rangle=0$. The action $S_{e-ph}$ describes electron-phonon interaction.

Let's consider $S_{e-ph}$ as a perturbation, and expand the exponent in it. Then, we average over phononic degrees of freedom assuming that $S_{e-ph}$ is linear in $a$ and $\bar{a}$, and thus all odd powers of $S_{e-ph}$ are zero. In such a way, we obtain
\begin{eqnarray}
\mathcal{Z}=\int \mathcal{D}[\bar{\psi},\psi,\bar{a},a]\exp{[i(S_{ph}+S_e)]}(1+iS_{e-ph}-\frac{1}{2}S_{e-ph}S_{e-ph}+...)=\int \mathcal{D}[\bar{\psi},\psi]\exp{[iS_e-\frac{1}{2}\langle S_{e-ph}S_{e-ph}\rangle]}.\ \ \
\end{eqnarray}
The last term is the action for electron-electron interaction mediated by phonons $S_{e-e}=i\langle S_{e-ph}S_{e-ph}\rangle/2$.

Now we consider the phonon action in detail. We assume that there are two phonon bands $\omega_{1,{\bf q}}$ and $\omega_{2,{\bf q}}$ that interact with the off-diagonal element $\delta_{\omega,{\bf q}}$:
\begin{eqnarray}
S_{ph}=\int \begin{pmatrix}\bar{a}_{1,\omega,{\bf q}} & \bar{a}_{2,\omega,{\bf q}}\end{pmatrix}\begin{pmatrix}\omega-\omega_{1,{\bf q}} & -\delta_{\omega,{\bf q}}\\ -\delta_{\omega,{\bf q}} & \omega-\omega_{2,{\bf q}}\end{pmatrix}\begin{pmatrix}a_{1,\omega,{\bf q}} \\ a_{2,\omega,{\bf q}}\end{pmatrix}d\omega d{\bf q}.
\end{eqnarray}
The electron-phonon interaction is described as
\begin{eqnarray}
S_{e-ph}=\int d\omega d{\bf q}F_{\omega,{\bf q}}\sum_{j=1,2}(a_{j,\omega,{\bf q}}+\bar{a}_{j,-\omega,-{\bf q}})\int d\Xi d{\bf K}\bar{\psi}_{\omega+\Xi,{\bf K}+{\bf q}}\psi_{\Xi,{\bf K}}=\int d\omega d{\bf q}F_{\omega,{\bf q}}\sum_{j=1,2}(a_{j,\omega,{\bf q}}+\bar{a}_{j,-\omega,-{\bf q}})\rho_{\omega,{\bf q}}.\ \ \ \ \ \ \ \ \ 
\end{eqnarray}
Here, $F_{\omega,{\bf q}}$ is the prefactor that usually contains material characteristics and constants, for instance, elastic constants. This gives
\begin{eqnarray}
\langle S_{e-ph}S_{e-ph}\rangle=\int \mathcal{D}[\bar{a},a]e^{iS_{ph}}S_{e-ph}S_{e-ph}=\\ \nonumber=\int d{\bf q}d{\bf q'}d\omega d\omega'F_{\omega,{\bf q}}F_{\omega',{\bf q'}}\rho_{\omega,{\bf q}}\rho_{\omega',{\bf q'}}\int \mathcal{D}[\bar{a},a]\sum_{i,j=1,2}[a_{i,\omega,{\bf q}}\bar{a}_{j,-\omega',-{\bf q}'}+\bar{a}_{i,-\omega,-{\bf q}}a_{j,\omega',{\bf q}'}] e^{iS_{ph}}.
\end{eqnarray}
Now we will use the property that
\begin{eqnarray}
\langle\bar{a}_{\bf q_1}a_{\bf q_2}\rangle=\int \mathcal{D}[\bar{a},a]\bar{a}_{\bf q_1}a_{\bf q_2}e^{-\sum_q\bar{a}_{\bf q}A_{\bf q}a_{\bf q}}=\frac{\delta_{{\bf q}_1,{\bf q}_2}}{A_{{\bf q}_1}},
\end{eqnarray}
in order to perform averaging. After averaging, we obtain
\begin{eqnarray}
\langle S_{e-ph}S_{e-ph}\rangle&=&i\int d{\bf q}d\omega F_{\omega,{\bf q}}F_{-\omega,-{\bf q}}\rho_{\omega,\bf q}\rho_{-\omega,-{\bf q}}\times \\ \nonumber&&\left[\frac{\omega_{1,{\bf q}}+\omega_{2,{\bf q}}-2\omega-2\delta_{\omega,{\bf q}}}{\delta_{\omega,{\bf q}}^2-(\omega-\omega_{1,{\bf q}})(\omega-\omega_{2,{\bf q}})}+\frac{\omega_{1,-{\bf q}}+\omega_{2,-{\bf q}}+2\omega-2\delta_{-\omega,-{\bf q}}}{\delta_{-\omega,-{\bf q}}^2-(\omega+\omega_{1,-{\bf q}})(\omega+\omega_{2,-{\bf q}})}\right].\ \ \ 
\end{eqnarray}
This yields the expression for the electron-electron interaction potential that we state in the main text in Eq. (17):
\begin{eqnarray}
V(\omega,{\bf q})=-\frac{F_{\omega,{\bf q}}F_{-\omega,-{\bf q}}}{2}\left[\frac{\omega_{1,{\bf q}}+\omega_{2,{\bf q}}-2\omega-2\delta_{\omega,{\bf q}}}{\delta_{\omega,{\bf q}}^2-(\omega-\omega_{1,{\bf q}})(\omega-\omega_{2,{\bf q}})}+\frac{\omega_{1,-{\bf q}}+\omega_{2,-{\bf q}}+2\omega-2\delta_{-\omega,-{\bf q}}}{\delta_{-\omega,-{\bf q}}^2-(\omega+\omega_{1,-{\bf q}})(\omega+\omega_{2,-{\bf q}})}\right].
\end{eqnarray}
Now let's assume that $\omega_{1,\pm{\bf q}},\omega_{2,\pm{\bf q}}\ll \delta_{\omega,{\bf q}},\delta_{-\omega,-{\bf q}},\omega$. Then we obtain
\begin{eqnarray}
V(\omega,{\bf q})\simeq F_{\omega,{\bf q}}F_{-\omega,-{\bf q}}\left[\frac{1}{\delta_{\omega,{\bf q}}-\omega}+\frac{1}{\delta_{-\omega,-{\bf q}}+\omega}\right]=F_{\omega,{\bf q}}F_{-\omega,-{\bf q}}\left[\frac{\delta_{-\omega,-{\bf q}}+\delta_{\omega,{\bf q}}}{(\delta_{\omega,{\bf q}}-\omega)(\delta_{-\omega,-{\bf q}}+\omega)}\right].
\end{eqnarray}
If the phonon interaction is strongly asymmetric in momentum space, $\delta_{\pm\omega,{\bf q}},\omega\gg\delta_{\pm\omega,-{\bf q}}$, then
\begin{eqnarray}
V(\omega,{\bf q})&\simeq& F_{\omega,{\bf q}}F_{-\omega,-{\bf q}}\frac{\delta_{\omega,{\bf q}}}{\omega(\delta_{\omega,{\bf q}}-\omega)},\\
V(\omega,-{\bf q})&\simeq& F_{\omega,{\bf q}}F_{-\omega,-{\bf q}}\frac{\delta_{-\omega,{\bf q}}}{-\omega(\delta_{-\omega,{\bf q}}+\omega)}.
\end{eqnarray}
If we further assume, $\delta_{\pm\omega,{\bf q}}\gg \omega$, we obtain Eq. (18) from the main text,
\begin{eqnarray}
V(\omega,{\bf q})\simeq -V(\omega,-{\bf q})\simeq\frac{F_{\omega,{\bf q}}F_{-\omega,-{\bf q}}}{\omega}.
\end{eqnarray}
Thus we obtain odd or asymmetric electron-electron interaction potential due to an externally-induced asymmetry in the phonon system, namely, the phonon-phonon interaction. The band structure of phonons, $\omega_{1,{\bf q}}$ and $\omega_{2,{\bf q}}$, can also be asymmetric, but here we have not used this property.




\end{widetext}

