We consider a mode II rupture which propagates along a planar main fault and encounters an intersection with a branching fault. Using an elastodynamic boundary integral equation formulation, allowing the failure path to be dynamically self-chosen, we study the following questions: Does the rupture initiate along the branch? Does it continue? Is the extensional or compressional side most favored for branching? Does rupture continue on the main fault too? Failure is described by a slip-weakening law for which the strength at any amount of slip is proportional to normal stress. Our results show that dynamic stresses around the rupture tip, which increase with rupture velocity at locations off the main fault plane relative to those on it, could initiate rupture on a branching fault. As suggested by prior work, whether branched rupture can be continued to a larger scale depends on principal stress directions in the prestress state and on rupture velocity. The most favored side for branching rupture switches from the extensional to the compressional side as we consider progressively shallower angles of the direction of maximum compressive prestress with the main fault. Simultaneous rupturing on both faults can be activated when the branching angle is wide but is usually difficult for a narrow branching angle due to strong stress interactions between faults. However, it can be also be activated by enhanced dynamic stressing when the rupture velocity is very near the Rayleigh velocity. Natural examples seem consistent with the simulations that we present.