The increased uptake of solar energy in the energy transition towards decarbonization has caused the issue of solar photovoltaic (PV) curtailments, resulting in significant economic losses and hindering the energy transition. To overcome this issue, battery energy storage systems (BESS) can serve as onsite backup sources for solar farms. However, the backup role of the BESS significantly limits its economic value, disincentivizing the BESS deployment due to high investment costs. Hence, it is essential to effectively reduce solar curtailment while ensuring viable operations of the BESS. To better understand the synergy of a co-located solar-BESS system in the real-time electricity market, we model the cooperative bidding processes of the solar farm and the BESS as a Markov game. We use a multi-agent deep reinforcement learning (MADRL) algorithm, known as multi-agent deep deterministic policy gradient, to concurrently maximize the overall revenue from the electricity market and reduce solar curtailments. We validate our MADRL-based strategy using data from a realistic solar farm operating in the Australian electricity market. The simulation results show that our MADRL-based coordinated bidding strategy outperforms both optimization-based and DRL-based benchmarks, generating higher revenue for the BESS and reducing more solar curtailments. Our work highlights the importance of coordination between the BESS and renewable generations for both economic benefits and progress towards net-zero transitions.