External field effects on the hydrodynamic instability of a nematic liquid crystal (LC) monodomain are analyzed using the two-dimensional Ericksen-Leslie continuum theory neglecting the Frank elasticity. It is found that in a static magnetic field, the so-called aligning-type LC with the third Leslie coefficient alpha(3) < 0 can, for an appropriate field strength and direction, exhibit tumbling under a simple shear flow, whereas the original tumbling-type LC with alpha(3) > 0 can show aligning in a strong magnetic field. The aligning-tumbling and tumbling-aligning transition points are calculated. Corresponding state diagrams for director tumbling and shear aligning are presented, and the tumbling loop is found. The aligning angles in the stable state and the tumbling periods in the unstable state are described analytically, along with the necessary and sufficient conditions for director tumbling. It is especially noteworthy that the direction of the external field plays a crucial role in controlling the hydrodynamic instability of LCs. As a consequence, this paper affords an interesting and feasible model system of a complex anisotropic fluid and establishes a framework for later theoretical and experimental studies.