> Where my_thread is a pointer to the current Fil-C thread, which Fil-C passes around as the first argument in all calls.

Does this just mean you reserve a register for the current thread? In which case you could explain it as a reserved register (like FS used for TLS). Describing it as "passes around as the first argument in all calls" makes it sound inefficient–but whether it actually is depends on how you implement it.