Contact-rich tasks continue to present many challenges for robotic manipulation. In this work, we leverage a multimodal visuotactile sensor within the framework of imitation learning (IL) to perform contact-rich tasks that involve relative motion (e.g., slipping and sliding) between the end-effector and the manipulated object. We introduce two algorithmic contributions, tactile force matching and learned mode switching, as complimentary methods for improving IL. Tactile force matching enhances kinesthetic teaching by reading approximate forces during the demonstration and generating an adapted robot trajectory that recreates the recorded forces. Learned mode switching uses IL to couple visual and tactile sensor modes with the learned motion policy, simplifying the transition from reaching to contacting. We perform robotic manipulation experiments on four door-opening tasks with a variety of observation and algorithm configurations to study the utility of multimodal visuotactile sensing and our proposed improvements. Our results show that the inclusion of force matching raises average policy success rates by 62.5%, visuotactile mode switching by 30.3%, and visuotactile data as a policy input by 42.5%, emphasizing the value of see-through tactile sensing for IL, both for data collection to allow force matching, and for policy execution to enable accurate task feedback. Project site: https://papers.starslab.ca/sts-il/