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Political polarization represents a rising issue in many countries, making it more and more important to understand its relation to cognitive-motivational and social influence mechanisms. Yet, the link between micro-level mechanisms and macro-level phenomena remains unclear. We investigated the consequences of individuals striving for cognitive coherence in their belief systems on political polarization in society in an agent-based model. In this, we formalized how cognitive coherence affects how individuals update their beliefs following social influence and self-reflection processes. We derive agents' political beliefs as well as their subjective belief systems, defining what determines coherence for different individuals, from European Social Survey data via correlational class analysis. The simulation shows that agents polarize in their beliefs when they have a strong strive for cognitive coherence, and especially when they have structurally different belief systems. In a mathematical analysis, we not only explain the main findings but also underscore the necessity of simulations for understanding the complex dynamics of socially embedded phenomena such as political polarization.