In this presentation, a framework for coupling arbitrary Lagrangian-Eulerian fluid-structure interaction with phase-field fracture is suggested. The key idea is based on applying the weak form of phase-field fracture, including a crack irreversibility constraint, to the nonlinear coupled system of Navier-Stokes and elasticity. The resulting setting is formulated via variational-monolithic coupling and has four unknowns: velocities, displacements, pressure, and a phase-field variable. The inequality constraint is imposed through penalization using an augmented Lagrangian algorithm. Temporal discretization is based on A-stable schemes and spatial discretization is realized with Galerkin finite elements. The nonlinear problem is solved with Newton's method. The framework is tested in terms of numerical examples in which computational stability is demonstrated by evaluating goal functionals on different spatial meshes and different time step sizes.