Virtual Oscillator (VO) control is the latest and promising control technique for grid-forming and grid-supporting inverters. VO Controllers (VOCs) provide time-domain synchronization with a connected electrical network. At the same time, a VOC can incorporate additional nested control loops for meets the system-level requirements such as fault ride-through capability. However, existing VOCs have two limitations. Firstly, the existing VOC does not have decoupled control over individual phases. As a result, the performance of the existing VOC is not satisfactory in presence of unbalanced grid voltages. Secondly, the fault ride-through performance of the existing VOC under an unbalanced fault is not satisfactory. The power delivery at a healthy phase gets affected by the faulty phase. This paper has introduced a modified VO based system-level controller to overcome the limitations mentioned above. Systematic development of the proposed control architecture with analytical reasoning is presented. Simulation studies and hardware experiments are conducted for validation.