The problem of blind detection in a dispersive multiple-input multiple-output (MIMO) code division multiple access (CDMA) channel is considered in this paper. Unlike previous studies, each user is assigned one spreading code to be employed on all of the transmit antennas, which poses a problem of data re-association at the receiver-end in the absence of prior information as to the channel state. Focusing on the differential Alamouti scheme, we propose a two-stage receive structure. The first stage performs a linear interference-blocking transformation, which allows user separation and inter-symbol interference (ISI) suppression. The second stage is, instead, a novel differential space-time block (STB) decoder suitable for frequency-selective channels. Interestingly, the proposed detector allows decoupling of the decisions on the transmitted symbols, while its blind implementation only requires a cubic (in the processing gain) complexity. A thorough performance assessment is undertaken to investigate, on one hand, the capability of acquiring the missing information, such as the system and the encoder timings, on the other hand, the interplay between the diversity gain provided by the MIMO structure of the communication system, and the additional co-channel interference that multiple transmit antennas produce in multi-path, multiple-access channels.