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This paper investigates communication over multiple-input multiple-output Rayleigh-product channels in the presence of both co-channel interference and thermal noise. We first present exact expressions for the marginal ordered eigenvalue distributions of the channel Gram matrix when multichannel beamforming is employed, from which we obtain exact results on the outage probability on each eigenmode. Our results are applicable to an extensive class of multichannel systems with generic system configurations. For particular scenarios of single-input multiple-output, keyhole, and multiple-input single-output channels, simplified closed-form expressions for the asymptotic distributions of the non-zero eigenvalue of the channel Gram matrix are derived. These results indicate that the diversity order is only determined by the second largest value among the number of transmit antennas, scatterers, and receive antennas. For these scenarios, we also derive concise closed-form expressions for the moments of the output signal-to-interference-plus-noise ratio, from which an explicit relationship to the moments in Rayleigh fading channels is revealed. These results are also used to investigate the impact of the number of channel scatterers on the bandwidth requirements for a given transmission rate and power, at low signal-to-noise ratios.