{h}^\dag_{{\bf k}, \alpha}  \Big| 0 \Big\rangle.\end{aligned}$$

 Methods and results. {#sec:mandr}
=====================

We calculate the hole spectral function $A_{\alpha}({\bf k}, \omega)$ using the self-consistent Born approximation (SCBA), see Ref. [@Martinez1991]. Such approach has been widely-successful in obtaining the cuprate spectral functions [@Martinez1991; @Ramsak1998; @Manousakis2007; @Wang2015; @Bieniasz2018] and amounts to neglecting the so-called crossing diagrams and summing all the other (‘rainbow diagrams’) to infinite order. The resulting self-consistent expressions for the self-energies and the Green’s function are given in Sec. \[sec:appendix\]. These equations are then solved numerically on a finite lattice of $36 \times 36$ ${\bf k}$ points. The resulting orbitally-integrated hole spectral function $A({\bf k}, \omega)$ is calculated for the realistic parameters of the model (see above) and is shown in Fig. \[result\](b).

The calculated hole spectral function qualitatively reproduces the incoherent and almost momentum-independent spectrum observed in the high binding energy part of Ca$_2$RuO$_4$ photoemission results found in Ref.  and reproduced in Fig. \[result\](a). Although the onset of several horizontal ‘stripes’ in the theoretical spectrum (see below) make the similarities between the theoretical and experimental spectral functions less apparent at first sight \[Fig. \[result\](b)\], a convolution of the theoretical spectral function with the available experimental resolution of ca. $0.5t$ yields 