The resistive incompressible magnetohydrodynamics equations are a difficult set of equations to solve numerically and, even more so, to conserve the appropriate conserved quantities, e.g., incompressibility or total energy. For this, methods have been created that use compatible spaces that conserve these quantities [1] (or a subset of them [2]). However, these methods have, till now, only used tensor product meshes without any adaptive refinement. This limits the size and/or difficulty of the problems that can be solved in a reasonable time. For this purpose, we will introduce an adaptive refinement method for the resistive incompressible MHD equations using THB-splines. We will use our previously introduced p-box meshes [3], which synergize well in terms of convergence, compatible spaces, and efficiency. For convergence of the adaptive method, it is known that the mesh must be of an admissibility class $c < \infty$. For this, we have refinement algorithms that ensure that a p-box mesh remains of the chosen admissibility classes after refinement/coarsening. For compatible spaces, in the case of a simple domain without any holes, the THB-spline spaces must form an exact complex. By the recent work of [4], p-boxes meshes offer a simple approach for constructing exact THB-spline complexes. Lastly, for efficiency, we can use the Bezier projector for THB-splines [3,5] to project intermediate solutions, which speeds up the non-linear solvers. In this talk, we present our adaptive numerical method and apply it to standard magnetohydrodynamics benchmarks. [1] E. S. Gawlik and F. Gay-Balmaz, “A finite element method for MHD that preserves energy, cross-helicity, magnetic helicity, incompressibility, and div B = 0,” Journal of Computational Physics, vol. 450, p. 110847, 2022. [2] Y. Zhang, A. Palha, A. Brugnoli, D. Toshniwal, and M. Gerritsma, “Decoupled structure-preserving discretization of incompressible MHD equations with general boundary conditions,” 2024. [3] K. W. Dijkstra, D. Toshniwal and C. Giannelli “Macro-element Refinement schemes for Bezier projection with adaptive Truncated Hierarchical B-splines”, in preparation. [4] K. Shepherd and D. Toshniwal, “Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in Rn”, Found Comput Math, 2024. [5] K. W. Dijkstra and D. Toshniwal, “A Characterization of Linear Independence of THB-Splines in Rn and Application to B´ezier Projection,” Approximation Theory