Energy management system (EMS) and volt-var optimization (VVO) programs are jointly employed to achieve the efficient operation of active distribution networks (ADNs). However, current studies often overlook that the operation of ADNs is coupled with trading energy in real-time (RT) energy markets. To fill this gap, this paper develops an adaptive robust optimization (ARO) model of ADNs in RT energy markets. The studied ADN consists of photovoltaic (PV) units, electric vehicle (EV) charging stations, and flexible demands interconnected through a medium-voltage distribution network. Through the proposed model, the integrated EMS and VVO programs can decide to offer energy in the hourly RT energy market, schedule flexible demands, and adjust slow-acting devices (e.g., on-load tap changers and capacitors), while anticipating the operation conditions of the ADN under uncertain prices, charging power level of EVs, and available PV power generation through the uncertainty sets. Once the market outcomes are known, the integrated EMS and VVO programs dispatch the fast-acting devices of PV units and EV charging stations. The results of a realistic case study indicate that prioritizing the reduction of voltage violations leads to lower energy trading in the RT energy market and a decrease in profit.