This paper presents an efficient GPU-accelerated global fit pipeline for LISA data analysis, called "Erebor," designed to analyze gravitational wave signals in the LISA data stream. The pipeline is capable of analyzing current state-of-the-art datasets and will grow as more components of the pipeline are developed. The algorithm is based on a Bayesian approach using Reversible-Jump Markov Chain Monte Carlo (RJMCMC) techniques to characterize the uncertain number of Galactic Binaries (GBs) in the data. It also employs the concept of the global fit "wheel," where separate modules run in parallel and communicate to perform the complex analysis more efficiently. The pipeline uses an Ensemble Sampling setup for MCMC runs, allowing for more efficient marginalization over model types. It also employs the "group stretch" MCMC proposal, which requires an ensemble setup and minimal tuning. The algorithm is optimized for GPU usage, taking advantage of their computational and energy efficiency. The pipeline is tested on the LDC2A "Sangria" dataset, which contains simulated LISA data with signals from Massive Black Hole Binaries (MBHBs), GBs, and a parameterized noise spectrum. The pipeline successfully recovers posterior distributions for all 15 (6) injected MBHBs in the training (hidden) dataset and catalogs ~12,000 GB sources (~8,000 with high confidence). The pipeline includes a detailed description of gravitational-wave signals and likelihood computations for MBHB and GB signals, a parameterized model for noise and foreground confusion, and an overview of MCMC methods. It also describes the initial aspects of the global fit search, the full global fit parameter estimation algorithm, and the results of the LDC2A analysis. The algorithm is designed to handle the complexity of the LISA data stream, including non-stationary noise effects and the need to include the orbit and performance of the LISA experiment in any global analysis. The pipeline is expected to be a large developmental project requiring many participating groups and expertise. The algorithm is designed to be efficient and scalable, with the potential to be used in future LISA data analysis efforts.This paper presents an efficient GPU-accelerated global fit pipeline for LISA data analysis, called "Erebor," designed to analyze gravitational wave signals in the LISA data stream. The pipeline is capable of analyzing current state-of-the-art datasets and will grow as more components of the pipeline are developed. The algorithm is based on a Bayesian approach using Reversible-Jump Markov Chain Monte Carlo (RJMCMC) techniques to characterize the uncertain number of Galactic Binaries (GBs) in the data. It also employs the concept of the global fit "wheel," where separate modules run in parallel and communicate to perform the complex analysis more efficiently. The pipeline uses an Ensemble Sampling setup for MCMC runs, allowing for more efficient marginalization over model types. It also employs the "group stretch" MCMC proposal, which requires an ensemble setup and minimal tuning. The algorithm is optimized for GPU usage, taking advantage of their computational and energy efficiency. The pipeline is tested on the LDC2A "Sangria" dataset, which contains simulated LISA data with signals from Massive Black Hole Binaries (MBHBs), GBs, and a parameterized noise spectrum. The pipeline successfully recovers posterior distributions for all 15 (6) injected MBHBs in the training (hidden) dataset and catalogs ~12,000 GB sources (~8,000 with high confidence). The pipeline includes a detailed description of gravitational-wave signals and likelihood computations for MBHB and GB signals, a parameterized model for noise and foreground confusion, and an overview of MCMC methods. It also describes the initial aspects of the global fit search, the full global fit parameter estimation algorithm, and the results of the LDC2A analysis. The algorithm is designed to handle the complexity of the LISA data stream, including non-stationary noise effects and the need to include the orbit and performance of the LISA experiment in any global analysis. The pipeline is expected to be a large developmental project requiring many participating groups and expertise. The algorithm is designed to be efficient and scalable, with the potential to be used in future LISA data analysis efforts.