General relativity works in all reference frames, so inertial forces are real in it. And due to the equivalence principle, gravity should be also considered inertial. So what is a good term for the other forces?
In general relativity, "inertial forces" (including gravity) are automatically accounted for by the geodesic equation (which in some sense replaces/generalizes Newton's First Law). They are considered simply part of the geometry spacetime and are not normally considered "forces" at all. This frees up the term "forces" for external influences that enter as a "right handed side" for the geodesic equation (i.e. what you referred to as "other forces").