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Electron-hole bound pairs, or excitons, are common excitations in semiconductors. They can spontaneously form and ``condense'' into a new insulating ground state -- the so-called excitonic insulator -- when the energy of electron-hole Coulomb attraction exceeds the band gap. In the presence of electron-phonon coupling, a periodic lattice distortion often concomitantly occurs with this exciton condensation. However, similar structural transition can also be induced by electron-phonon coupling itself, therefore hindering the clean identification of bulk excitonic insulators based on reductionistic reasoning (e.g. which instability is the ``driving force'' of the phase transition). Using high-resolution synchrotron x-ray diffraction and angle-resolved photoemission spectroscopy techniques, we identify key electron-phonon coupling effects in a leading excitonic insulator candidate Ta2NiSe5. These include an extensive unidirectional lattice fluctuation and an electronic pseudogap in the normal state, as well as a negative electronic compressibility in the charge-doped broken-symmetry state. In combination with first principles and model calculations, we determine a minimal lattice model and the corresponding interaction parameters that capture the experimental observations. More importantly, we show how the Coulomb and electron-phonon coupling effects can be separated on the level of lattice model, and demonstrate a general framework beyond the reductionist approach in the investigation of correlated systems with intertwined orders.