Decomposing the electric field (E) into the contributions from generalized Ohm's law provides key insight into both nonlinear and dissipative dynamics across the full range of scales within a plasma. Using high-resolution, multispacecraft measurements of three intervals in Earth's magnetosheath from the Magnetospheric Multiscale mission, the influence of the magnetohydrodynamic, Hall, electron pressure, and electron inertia terms from Ohm's law, as well as the impact of a finite electron mass, on the turbulent E spectrum are examined observationally for the first time. The magnetohydrodynamic, Hall, and electron pressure terms are the dominant contributions to E over the accessible length scales, which extend to scales smaller than the electron gyroradius at the greatest extent, with the Hall and electron pressure terms dominating at sub-ion scales. The strength of the nonideal electron pressure contribution is stronger than expected from linear kinetic Alfven waves and a partial antialignment with the Hall electric field is present, linked to the relative importance of electron diamagnetic currents in the turbulence. The relative contribution of linear and nonlinear electric fields scale with the turbulent fluctuation amplitude, with nonlinear contributions playing the dominant role in shaping E for the intervals examined in this study. Overall, the sum of the Ohm's law terms and measured E agree to within similar to 20% across the observable scales. These results both confirm general expectations about the behavior of E in turbulent plasmas and highlight features that should be explored further theoretically.