Although it is not possible to have a covariant treatment of inertial forces in general, spinor fields we can define specific objects that describe forces of inertial type but which are real tensors: see https://link.springer.com/article/10.1140/epjc/s10052-018-6260-8 for more details. This surprised me a bit, so I'd like to have a discussion about the subject with whomever is interested.