Lets say we have a non-interacting translation invariant Hamiltonian $H$ in a matrix form:
$$H=\Psi^{\dagger}\bar{H}\Psi$$
where $\Psi^{\dagger}=(\Psi_{\uparrow}^{\dagger}\quad \Psi_{\downarrow}^{\dagger} \quad\Psi_{\downarrow}\quad-\Psi_{\uparrow})$ is the extended nambu spinor.
In such a basis what is the meaning of a particular element of the matrix representation of the retarded green function:
$$\bar{G}^{R}(\omega)=\frac{1}{\omega-\bar{H}+i\delta}$$
what is $\bar{G}^{R}_{ij}$?
As a general recipe, the retarded Green's function is defined as $$ G^{R}_{ij}(t,t') = -i \theta(t,t') \langle \{\psi_{i}, \psi_{j}^{\dagger} \} \rangle, $$ where $\theta$ is the step-function, and $\{ ,\}$ denotes anticommutator. $\psi_{k}$ based on your suggested language is k-th elemnt of $\Psi$. For instance, $\psi_{1}= \Psi_{\uparrow}$. The full retarded Green's function would be $$ G^{R}(t,t')= -i \theta(t,t') \begin{pmatrix} \langle \{ \Psi_{\uparrow}(t) , \Psi^{\dagger}_{\uparrow}(t') \} \rangle& \langle \{ \Psi_{\uparrow}(t), \Psi^{\dagger}_{\downarrow}(t') \} \rangle& \langle \{ \Psi_{\uparrow}(t), \Psi_{\downarrow}(t') \} \rangle & -\langle \{ \Psi_{\uparrow}(t), \Psi_{\uparrow}(t') \} \rangle\\ \langle \{ \Psi_{\downarrow}(t) , \Psi_{\uparrow}^{\dagger}(t') \} \rangle & \langle \{ \Psi_{\downarrow}(t) , \Psi_{\downarrow}^{\dagger}(t') \}\rangle & \langle \{ \Psi_{\downarrow}(t) , \Psi_{\downarrow}(t') \} \rangle & -\langle \{ \Psi_{\downarrow}(t) ,\Psi_{\uparrow}(t') \} \rangle \\ \langle \{ \Psi^{\dagger}_{\downarrow}(t) , \Psi_{\uparrow}^{\dagger}(t') \} \rangle & \langle \{ \Psi^{\dagger}_{\downarrow}(t) , \Psi_{\downarrow}^{\dagger}(t') \} \rangle& \langle \{ \Psi^{\dagger}_{\downarrow}(t) , \Psi_{\downarrow}(t') \}\rangle & -\langle \{ \Psi^{\dagger}_{\downarrow}(t) ,\Psi_{\uparrow}(t') \} \rangle\\ -\langle \{ \Psi^{\dagger}_{\uparrow}(t) , \Psi_{\uparrow}^{\dagger}(t') \} \rangle& -\langle \{ \Psi^{\dagger}_{\uparrow}(t) , \Psi_{\downarrow}^{\dagger}(t') \} \rangle& -\langle \{ \Psi^{\dagger}_{\uparrow}(t) , \Psi_{\downarrow}(t') \} \rangle& \langle \{ \Psi^{\dagger}_{\uparrow}(t) ,\Psi_{\uparrow}(t') \} \rangle \end{pmatrix} $$
The suggested generic definition of the retarded Green's function in your question is just the Fourier transform of the above matrix in the basis where $\overline{H}$is defined.
