Surge is by definition a transient phenomenon. The compressor operating point can be forced momentarily across its surge line, but it cannot stay there (in most realistic situations). The airfoils stall (at least some of them), and pressure and flow drop precipitously, crossing back into a non-stalled, viable operating area. If no external adjustment is made to the boundary conditions, the operating point may follow the same trajectory repeatedly, i.e., increasing in pressure, crossing the surge line, stalling, and so on.
It is also possible for a compressor operating point to ride right on the surge line, but then it is just executing the cycle describe above on a small scale, where the situation is quasi-stable.
I can imagine an experiment with special apparatus that forces a compressor to be continuously on the wrong side of the surge line, but then it is continuously stalled (at least part of it) also.
If you look at a compressor map you will see that the surge line delineates an area of relatively high pressure ratio vs mass flow, from the stable operating area.
"Schiefgehen wird, was schiefgehen kann" - das Murphygesetz