I interpret the question in a bit more general terms, as a request to "shed light" on the phenomenon that "oftentimes many eigenvalues lie exactly on the real axis".
The key thing to notice is that the spectrum of the Lindbladian ${\cal L}$ is invariant under complex conjugation. So if we consider a parameter dependence ${\cal L}(t)$, an eigenvalue that is on the real axis for some $t=t_0$ cannot move into the complex plane when $t$ is increased slightly. It must first collide with another real eigenvalue, to become a complex conjugated pair.
This mechanism generically produces a number $N_{\rm real}$ of real eigenvalues that scales with the square root of the total number $N_{\rm tot}$ of eigenvalues. Precise calculations can be done in the Ginibre ensemble of random matrices, where $$\mathbb{E}[N_{\rm real}]=\sqrt{2N_{\rm tot}/\pi}+{\cal O}(N_{\rm tot}^{-1}),$$ but the same square root law is observed in other ensembles (see, for example, appendix A.4 of arXiv: 1305.2924 ).