We introduce a two-parameter ensemble of random discrete-time Markov models that simultaneously captures critical slowing down and broken detailed balance. Extending a previously studied heterogeneous Markov ensemble, we incorporate correlations between forward and backward transition rates through a single asymmetry parameter $\gamma$, while heterogeneity is controlled by $\epsilon$. Using results from random matrix theory, we identify a critical locus ${\epsilon }_c(\gamma ,N)$ at which relaxation times diverge and spectral universality breaks down, in Markov models with $N$ states. We characterize the behavior of entropy production, predictive information, and relaxation dynamics across the ensemble, showing that many observables depend strongly on heterogeneity but only weakly on asymmetry, except near the symmetric limit. Applying maximum-likelihood inference to human fMRI and EEG data, we find that both modalities operate near the predicted critical locus and occupy a similar region of the $\epsilon -\gamma$ plane, supporting a superuniversality of human brain dynamics. While ensemble averages are well captured by the null model, empirical data exhibit substantially enhanced variability, indicating subject-specific structure beyond random expectations. Our results unify criticality and nonequilibrium measures within a single framework and clarify their intertwined role in the analysis of complex biological dynamics.

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• Complex systems
• Network phase transitions
• Neuroscience, neural computation & artificial intelligence
• Markovian processes