We develop a theory of quantum fluctuations and squeezing in a three-dimensional Bose-Einstein condensate atom interferometer with nonlinear losses. We use stochastic equations in a truncated Wigner representation to treat quantum noise. Our approach includes the multi-mode spatial evolution of spinor components and describes the many-body dynamics of a mesoscopic quantum system. PACS numbers: 03.75. Gg, 03.75.Dg, 67.85.Fg, 67.85.De Atom interferometry is an important quantum technology at the heart of many proposed future applications of ultra-cold atomic physics. Bose-Einstein condensates (BECs) or atom lasers are macroscopic quantum objects and have potential advantages as interferometric detectors and sensors, provided one can precisely extract atomic phase information. However, unlike photons, atoms can interact strongly, causing dephasing and loss of interference fringes. An intimate understanding of quantum many-body dynamics is the key to calculating interaction-induced dephasing in the measurement process. This is essential for a quantitative theory of atom interferometry.In this Letter we present a simple, yet quantitatively accurate theoretical approach to simulating the dynamics and evaluating limits of atom interferometry at large atom number, using a truncated Wigner representation [1][2][3]. This method extends the conventional Gross-Pitaevskii equations describing a Bose condensate to include quantum noise effects, including noise due to linear and nonlinear losses. The theory allows the accurate inclusion of quantum fluctuations due to nonlinear losses, which is a dominant effect when atom numbers are increased to improve fringe visibility.Importantly, we can clearly demonstrate where fringe visibility is driven by quantum fluctuations, and where it is driven by trap inhomogeneity and dynamical effects, in order to choose optimal conditions for quantum noise reduction and spin squeezing. These calculations are a first step towards understanding mesoscopic superpositions and entanglement in ultra-cold atomic gases. An advantage of our method compared to the variational approaches used elsewhere [4,5] is that it allows us to treat a large number of independent field modes and particles, thus including degrees of freedom that are excited due to collisional and nonlinear loss dynamics [6,7]. Our theory can be readily extended to include finite temperature initial conditions [2,8], which will be treated elsewhere. Nonlinear losses and finite temperature effects can be also described within the confines of the variational approach [9,10].Quantum phase-diffusion is defined as the phase noise induced by number fluctuations which are conjugate to phase. This is a fundamental feature of BEC interferometry, and can only be removed when there are no interactions. However, there are other reasons for decoherence, which are also important. The approach used here captures all three significant features of atom interferometry that can result in decoherence: phase-diffusion, losses, and trap inhomogeneity effe...