We study a two-institution stable matching model in which candidates from two distinct groups are evaluated using partially correlated signals that are group-biased. This extends prior work (which assumes institutions evaluate candidates in an identical manner) to a more realistic setting in which institutions rely on overlapping, but independently processed, criteria. These evaluations could consist of a variety of informative tools such as standardized tests, shared recommendation systems, or AI-based assessments with local noise. Two key parameters govern evaluations: the bias parameter $β\in (0,1]$, which models systematic disadvantage faced by one group, and the correlation parameter $γ\in [0,1]$, which captures the alignment between institutional rankings. We study the representation ratio, i.e., the ratio of disadvantaged to advantaged candidates selected by the matching process in this setting. Focusing on a regime in which all candidates prefer the same institution, we characterize the large-market equilibrium and derive a closed-form expression for the resulting representation ratio. Prior work shows that when $γ= 1$, this ratio scales linearly with $β$. In contrast, we show that the representation ratio increases nonlinearly with $γ$ and even modest losses in correlation can cause sharp drops in the representation ratio. Our analysis identifies critical $γ$-thresholds where institutional selection behavior undergoes discrete transitions, and reveals structural conditions under which evaluator alignment or bias mitigation are most effective. Finally, we show how this framework and results enable interventions for fairness-aware design in decentralized selection systems.