We propose a multichannel version of quantum electro-optic sampling involving monochromatic field modes. It allows for multiple simultaneous measurements of arbitrarily many and field quadratures for a single quantum-state copy, while independently tuning the interaction strengths at each channel. In contrast to standard electro-optic sampling, the sampled midinfrared (MIR) mode undergoes a nonlinear interaction with multiple near-infrared pump beams. We present a complete positive operator-valued measure description for quantum states in the MIR mode. The probability distribution of the electro-optic signal outcomes is shown to be related to an -parametrized phase-space quasiprobability distribution of the indirectly measured MIR state, with the parameter depending solely on the quantities characterizing the nonlinear interaction. Furthermore, we show that the quasiprobability distributions for the sampled and postmeasurement states are related to each other through a renormalization and a change in the parametrization. This result is then used to demonstrate that two consecutive measurements of both and quadratures can outperform eight-port homodyne detection.