The tethered space net robot offers an effective solution for active space debris removal due to its large capture envelope. However, most existing studies overlook the evasive behavior of non-cooperative targets. To address this, we model an orbital pursuit–evasion game involving a tethered net and propose a game theory-based leader–follower tracking control strategy. In this framework, a virtual leader—defined as the geometric center of four followers—engages in a zero-sum game with the evader. An adaptive dynamic programming method is employed to handle input saturation and compute the Nash Equilibrium strategy. In the follower formation tracking phase, a synchronous distributed model predictive control approach is proposed to update all followers’ control simultaneously, ensuring accurate tracking while meeting safety constraints. The feasibility and stability of the proposed method are theoretically analyzed. Additionally, a body-fixed reference frame is introduced to reduce the capture angle. Simulation results show that the proposed strategy successfully captures the target and outperforms existing methods in both formation keeping and control efficiency.