The existing power network of a Mediterranean island is used as test bed for the study of grid-forming converter dynamics in weak grid conditions. The control scheme adopted for the converter falls in the category of the so called virtual synchronous machines (VSM). Particular attention is committed to the interaction between the converter controls and the synchronous machines dynamics. Time delays typically associated to measurements and filters are considered in the study, and included in the system model with a specific mathematical approach. Time-domain simulations show the occurrence of oscillatory instability in the system, with VSM and synchronous machines swinging against each other and progressively diverging. The modal analysis is performed for a deeper grasp of the system dynamics: the reasons of the instability are identified as the lack of damping in the system and the swinging interaction between grid-forming converter and synchronous machines. The opportunity of different corrective actions is analyzed and discussed, recognizing PSS-like control and active power phase compensation as the most suitable methods for the stabilization of the system.