Simulating deformable objects under rich interactions remains a fundamental challenge for real-to-sim robot manipulation, with dynamics jointly driven by environmental effects and robot actions. Existing simulators rely on predefined physics or data-driven dynamics without robot-conditioned control, limiting accuracy, stability, and generalization. This paper presents SoMA, a 3D Gaussian Splat simulator for soft-body manipulation. SoMA couples deformable dynamics, environmental forces, and robot joint actions in a unified latent neural space for end-to-end real-to-sim simulation. Modeling interactions over learned Gaussian splats enables controllable, stable long-horizon manipulation and generalization beyond observed trajectories without predefined physical models. SoMA improves resimulation accuracy and generalization on real-world robot manipulation by 20%, enabling stable simulation of complex tasks such as long-horizon cloth folding.