Intelligent reflecting surface (IRS) is a promising technology for the 6th generation of wireless systems, realizing the smart radio environment concept. In this paper, we present a novel tensor-based receiver for IRS-assisted multiple-input multiple-output communications capable of jointly estimating the channels and the transmitted data streams in a semi-blind fashion. Assuming a fully passive IRS architecture and introducing a simple space-time coding scheme at the transmitter, the received signal model can be advantageously built using the PARATUCK tensor model, which can be seen as a hybrid of parallel factor analysis and Tucker models. Exploiting the algebraic structure of the PARATUCK tensor model, a semi-blind receiver is derived. The proposed receiver is based on a trilinear alternating least squares method that iteratively estimates the two involved - IRS- base station and user terminal-IRS-communication channels and the transmitted symbol matrix. We discuss identifiability conditions that ensure the joint semi-blind recovery of the involved channel and symbol matrices, and propose a joint design of the coding and IRS reflection matrices to optimize the receiver performance. For the proposed semi-blind receiver, the derivation of the expected Cram\'er-Rao lower bound is also provided. A numerical performance evaluation of the proposed receiver design corroborates its superior performance in terms of the normalized mean squared error of the estimated channels and the achieved symbol error rate.