A multiple-input-multiple-output (MIMO) version of Costa's dirty paper channel is studied, where both the input signal and the state experience ergodic fading with channel state information at the receiver. An inner bound using lattice coding and decoding is derived and its gap to the point-to- point ergodic capacity is computed. Under Rayleigh fading, the gap is a constant that vanishes as the number of receive antennas increases, even at finite signal-to-noise ratio (SNR). This implies that under certain configurations the point-to- point capacity can be almost realized in the presence of fading dirt, similar to Costa's result for the non-fading scalar case. The proposed scheme is applied to a class of MIMO broadcast channels with receive channel state information. The scheme is compared to time sharing as well as a dirty paper scheme with non-causal channel state information at the transmitter. The lattice coding scheme is shown to outperform time sharing as well as achieve most of the region of the dirty paper scheme for the cases under study.