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This study investigates the complexity, efficiency, and sectoral interdependencies of the S&P Global BMI indices during critical global events, including the COVID-19 pandemic and the Russia–Ukraine war. The analysis is conducted in three dimensions: (1) evaluating market efficiency using permutation entropy and the Fisher information measure, (2) exploring sectoral alignments through clustering techniques (hierarchical and k-means clustering), and (3) assessing the influence of geopolitical risk using Multifractal Detrended Cross-Correlation Analysis (MFDCCA). The results highlight significant variations in informational efficiency across sectors, with Utilities and Consumer Staples exhibiting high efficiency, while Emerging Markets and Financials reflect lower efficiency levels. Temporal analysis reveals widespread efficiency declines during the pandemic, followed by mixed recovery patterns during the Ukraine conflict. Clustering analysis uncovers dynamic shifts in sectoral relationships, emphasizing the resilience of defensive sectors and the unique behavior of Developed BMI throughout crises. MFDCCA further demonstrates the multifractality in cross-correlations with geopolitical risk, with Consumer Staples and Energy showing stable persistence and Information Technology exhibiting sensitive complexity. These findings emphasize the adaptive nature of global markets in response to systemic and geopolitical shocks, offering insights for risk management and investment strategies. Full article