In this chapter, further extensions, algorithmic motivations and descriptions of a framework for coupling arbitrary Lagrangian-Eulerian fluid-structure interaction with phase-field fracture are addressed. This setting is formulated via variational-monolithic coupling and has four unknowns: velocities, displacements, pressure, and a phase-field variable. Moreover, the phase-field variable is subject to a crack irreversibility constraint. The resulting formulation is a highly nonlinear multiphysics variational inequality system. Here, the irreversibility constraint is imposed through penalization using an augmented Lagrangian algorithm. This formulation is treated with a variational-monolithic arbitrary Lagrangian-Eulerian approach,which allows one to use implicit time-stepping methods for a robust discretization in time. The first aim is a revisit of our concept to couple fluid-structure and phase-field fracture including lists of major challenges. Our second focus is on adaptive time-step control of goal functionals based on a heuristic estimator. Third, algorithmic details on Newton’s method for solving the nonlinear problem are provided. The proposed concepts are substantiated with numerical test cases.