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Using the effective field theory framework for extended objects we describe the evolution of spinning compact objects in the late inspiral of the coalescence of a binary, before the plunge and merger, by including leading order corrections due to spin, tides, dissipation and gravitational wave radiation. Our implementation is of particular relevance for probing the stellar structure of compact objects with gravitational wave observations. A spinning compact object in the effective field theory framework is described as a spinning point particle, with its finite size effects encoded in higher order operators in the effective action, operators which have coefficients that encapsulates the internal structure of the star. For the inspiral regime described by non-relativistic general relativity, post-Newtonian corrections to each term of the action can be obtained in a diagrammatic approach, including gravitational radiation effects. Taking into account the aforementioned ingredients of the effective theory, we solve for the dynamics of the inspiral of binary systems using an algorithm for point particle simulations. We extract the gravitational wave as a function of the orbital frequency, input that is generated numerically and then evaluated in the analytic function of the waveform. By performing illustrative numerical experiments of systems that the LIGO-Virgo observatories have already detected, we show the role of the stellar structure and its coefficients in the phase evolution of the waveform, as well as the order in which they arise and the sensitivity required for the gravitational wave observatories to measure them. If these coefficients are to be measured, tight constraints on fundamental physics can be placed.