Thermodynamic equilibrium can be defined as the terminal state of a macroscopic system when all external influences have been removed, and all the properties of the system have become time-independent. The macroscopic state of a system at thermodynamic equilibrium is characterized by a few physical quantities (thermodynamic variables) whose values do not depend on the system's history.
Mention to "external influences" is important to exclude stationary systems which do not correspond to the system at equilibrium (for example, a piece of metal with different temperatures at the extreme sides; in that case, it is possible to get a stationary state but in the presence of a flux of heat).
The apparently simple concept of thermodynamic equilibrium could not be simple to check in real systems. Feynman, at the beginning of his book on Statistical Mechanics, wrote that we have thermal equilibrium.
... if all the "fast" things have happened and all the "slow" things not.
The reason is that some very slow processes on the time scale of lab measurements may be in place and eventually modify the system's state over a geological scale.
Now, in the specific case of your system, it is a composite system. Assuming that the walls without strips are not only insulating but also perfectly rigid and not permeable to diffusion of particles, subsystem B, waiting long enough, will reach thermodynamic equilibrium independently of what happens elsewhere.
Subsystem A won't be at equilibrium until fluxes are in place. However, assuming that out of the diathermal wall, there is a thermostat, it could equilibrate with the thermostat sooner or later. At this point, the global system is at thermodynamic equilibrium until some change on external or internal conditions is made.
It is meaningless to speak about equilibrium of subsystem A with respect to subsystem B if subsystem B is fully isolated. In general temperature of A and B will differ but, due to the insulating wall, this is not hampering the thermodynamic equilibrium of the composite system. If at some time, the separation wall between A and B would be transformed from insulating to diathermal, at that point, the two subsystems won't be, in general, at thermal equilibrium, and the global system will evolve towards a state where the temperature of A and B will coincide.