Research on political belief systems increasingly relies on belief network analysis to infer ideological structure, cultural schemas, and polarization from survey data. The two dominant approaches, correlation-based and precision-based, are typically treated as imperfect but essentially direct maps of shared cognitive structure. In this paper, I argue that these reconstruction pipelines are grounded in fundamentally different causal models: correlation networks assume that associations between beliefs are generated by latent cultural schemas, while precision networks assume that beliefs directly constrain one another through conditional dependence. Using simulations that generate individual survey responses from latent schematic structures, population mixtures, and measurement noise, I evaluate how these pipelines reconstruct the same underlying belief systems. I find that correlation-based networks reliably recover modular structure and relative centrality when beliefs are produced by shared latent causes, while precision-based networks systematically distort that structure, fragmenting coherent clusters and producing spurious clusters, hubs, and bridges. These distortions persist even with large samples and low measurement error, indicating that they arise from generative model mis-specification rather than sampling variability or statistical noise. The results imply that influential claims about ideological constraint and polarization based on precision networks may be artifacts of the reconstruction pipeline rather than properties of cultural and political cognition.