For a convergent solution (boundary layer atmospheric flow); is it possible to determine real world separation?

From my understanding in the real world any separation is usually the result of a turbulent eddy, and is a transient phenomena, and hence will not converge. Only in special circumstances will separation occur where a stable eddy results, this would be similar to a standing wave.

Or have I got this all wrong?

Separation can occur for laminar flows, and hence a turbulent eddy is not required. Separation is also possible for a steady flow - go look at results for low Re cylinder flows. In the most basic sense, separation occurs when the kinetic energy of the flow is insufficient to overcome the adverse pressure gradient.

your apprehension is mostly true. the steadiness of a bubble or generally a vortex is in direct relation with the Reynolds number of flow and more importantly the adverse pressure gradient of geometry which is producing separation bubble. on the other word if you assume a constant case ( same geometry ) the factor which is dominant in steadiness of phenomenon is Re .