The recent release of the second Gravitational-Wave Transient Catalog (GWTC-2) has increased significantly the number of known GW events, enabling unprecedented constraints on formation models of compact binaries. One pressing question is to understand the fraction of binaries originating from different formation channels, such as isolated field formation versus dynamical formation in dense stellar clusters. In this paper, we combine the cosmic binary population synthesis suite and the cmc code for globular cluster evolution to create a mixture model for black hole binary formation under both formation scenarios. For the first time, these code bodies are combined self-consistently, with cmc itself employing cosmic to track stellar evolution. We then use a deep-learning enhanced hierarchical Bayesian analysis to continuously sample over and constrain the common envelope efficiency α assumed in cosmic, the initial cluster virial radius r v adopted in cmc, and the intrinsic mixture fraction f between each channel. Under specific assumptions about other uncertain aspects of isolated binary and globular cluster evolution, we report the median and 90% confidence interval of three physical parameters for the intrinsic population ... This simultaneous constraint agrees with observed properties of globular clusters in the Milky Way and is an important first step in the pathway toward learning the astrophysics of compact binary formation through GW observations. (ProQuest: … denotes formulae omitted.)