This paper addresses robust communication on a fading relay channel in which the relay is connected to the decoder via an out-of-band digital link of limited capacity. Both the source-to-relay and the source-to-destination links are subject to fading gains, which are generally unknown to the encoder prior to transmission. To overcome this impairment, a hybrid automatic retransmission request (HARQ) protocol is combined with multi-layer broadcast transmission, thus allowing for variable-rate decoding. Moreover, motivated by cloud radio access network applications, the relay operation is limited to compress-and-forward. The aim is maximizing the throughput performance as measured by the average number of successfully received bits per channel use, under either long-term static channel (LTSC) or short-term static channel (STSC) models. In order to opportunistically leverage better channel states based on the HARQ feedback from the decoder, an adaptive compression strategy at the relay is also proposed. Numerical results confirm the effectiveness of the proposed strategies.