Redistricting plays a central role in shaping how votes are translated into political power. While existing computational methods primarily aim to generate large ensembles of legally valid districting plans, they often neglect the strategic dynamics involved in the selection process. This oversight creates opportunities for partisan actors to cherry-pick maps that, while technically compliant, are politically advantageous. Simply satisfying formal constraints does not ensure fairness when the selection process itself can be manipulated. We propose Agentmandering, a framework that reimagines redistricting as a turn-based negotiation between two agents representing opposing political interests. Drawing inspiration from game-theoretic ideas, particularly the Choose-and-Freeze protocol, our method embeds strategic interaction into the redistricting process via large language model (LLM) agents. Agents alternate between selecting and freezing districts from a small set of candidate maps, gradually partitioning the state through constrained and interpretable choices. Evaluation on post-2020 U.S. Census data across all states shows that Agentmandering significantly reduces partisan bias and unfairness, while achieving 2 to 3 orders of magnitude lower variance than standard baselines. These results demonstrate both fairness and stability, especially in swing-state scenarios.