Physics-consistent theoretical studies on RIS-parametrized wireless channels use models from multiport-network theory (MNT) to capture mutual-coupling (MC) effects. However, in practice, RIS design and radio environment are partially or completely unknown. We fill a research gap on how to estimate the MNT model parameters in such experimentally relevant scenarios. Our technique efficiently combines closed-form and gradient-descent steps, and it can be applied to multi-bit-programmable RIS elements. We discuss inevitable (but operationally irrelevant) parameter ambiguities. We experimentally validate our technique in an unknown rich-scattering environment parametrized by eight 6-bit-programmable RIS elements of unknown design. We experimentally evaluate the performance of RIS configurations optimized with the estimated MNT model and an MC-unaware cascaded model. While the models differ in accuracy by up to 17 dB, the end-to-end performance differences are small.