A novel coding strategy is proposed for a broadcast setting with two transmitter (TX) antennas and two single-antenna receivers (RX). The strategy consists of using space-time block coding to send a common message (to be decoded by both RXs) across the two TX antennas, while each TX antenna also sends a private message to one of the RXs. The relative weight of the private and common messages from each TX antenna is tuned to maximize the instantaneous achievable sum-rate of the channel. Closed-form expressions for the optimal weight factors are derived. In terms of the generalized degrees of freedom (GDoF) metric, the new scheme is able to achieve the sum-GDoF with finite precision channel state information at the transmitter (CSIT) of the two user broadcast channel. Moreover, as opposed to the existing rate-splitting schemes, the proposed scheme yields instantaneous achievable rates that are independent of the channel phases. This property is instrumental for link adaptation when only magnitude CSIT is available. Our numerical results indeed demonstrate the superiority of the scheme for the 2-user setting in case of magnitude CSIT. Extension to a more general K-user scenario is briefly discussed.