We developed a four-electrode acquisition system to measure the complex conductivity tensor of tight and anisotropic shale cores using an accurate impedance meter. The complex conductivity tensor was obtained by inverting complex electrical potential data (amplitude and phase) acquired over a distributed array of electrodes. Inversion of the acquired data was performed with a Markov chain Monte Carlo (McMC) sampler that can explicitly take into account the nonuniqueness of the inverse problem. This approach was validated with direct measurements of impedance tensor eigenvalues with a single-component measurement technique. Consistent results were obtained using these two methods validating those obtained through the McMC sampling strategy. Both the in-phase and quadrature conductivities were anisotropic with the same anisotropic ratio in agreement with a recently developed petrophysical model based on volume averaging. The type of anisotropy of the Bakken and Haynesville core samples was found to be transverse isotropic, with the anisotropic ratio on the order of 10. In-phase conductivity was found to be independent of frequency up to 100 Hz, while the magnitude of the quadrature conductivity increased with frequency in the range from 0.1 to 45 kHz. Both components were sensitive to water saturation. Surface conductivity could be reliably predicted from the quadrature conductivity in agreement with the model prediction. The relationship between surface and quadrature conductivities was independent on saturation and anisotropy.