The paper presents an efficient GPU-accelerated multi-source global fit pipeline for analyzing LISA data, designed to handle the high dimensionality, multiple model types, and complex noise profile of gravitational wave signals. The pipeline, named "Erebor," is capable of analyzing current state-of-the-art datasets and can be extended with future improvements. The authors detail their global fit algorithm, which uses Reversible-Jump Markov Chain Monte Carlo (RJMCMC) techniques to fit for and characterize the number of Galactic Binaries (GBs) and Massive Black Hole Binaries (MBHBs) in the data. They describe the pipeline strategy, algorithmic setup, and results from analyzing the LDC2A Sangria dataset, which includes Massive Black Hole Binaries, compact Galactic Binaries, and a parameterized noise spectrum. The pipeline recovers posterior distributions for all 15 (6) injected MBHBs in the training (hidden) dataset and catalogs approximately 12,000 GB sources, with about 8,000 being high-confidence detections. The sources and their posterior distributions are provided in publicly available catalogs. The paper also discusses the gravitational-wave signal likelihoods, input template models, noise treatment, and Markov Chain Monte Carlo methods used in the pipeline.The paper presents an efficient GPU-accelerated multi-source global fit pipeline for analyzing LISA data, designed to handle the high dimensionality, multiple model types, and complex noise profile of gravitational wave signals. The pipeline, named "Erebor," is capable of analyzing current state-of-the-art datasets and can be extended with future improvements. The authors detail their global fit algorithm, which uses Reversible-Jump Markov Chain Monte Carlo (RJMCMC) techniques to fit for and characterize the number of Galactic Binaries (GBs) and Massive Black Hole Binaries (MBHBs) in the data. They describe the pipeline strategy, algorithmic setup, and results from analyzing the LDC2A Sangria dataset, which includes Massive Black Hole Binaries, compact Galactic Binaries, and a parameterized noise spectrum. The pipeline recovers posterior distributions for all 15 (6) injected MBHBs in the training (hidden) dataset and catalogs approximately 12,000 GB sources, with about 8,000 being high-confidence detections. The sources and their posterior distributions are provided in publicly available catalogs. The paper also discusses the gravitational-wave signal likelihoods, input template models, noise treatment, and Markov Chain Monte Carlo methods used in the pipeline.