The magnetic compass of birds is embedded in the visual system and it has been hypothesizedthat the primary sensory mechanism is based on a radical pair reaction. Previous models of magnetoreception have assumed that the radical pair-forming molecules are rigidly fixed in space, and this assumption has been a major objection to the suggested hypothesis. In this article, we investigate theoretically how much disorder is permitted for the radical pair-forming, protein-based magnetic compass in the eye to remain functional. Our study shows that only one rotational degree of freedom of the radical pair-forming protein needs to be partially constrained, while the other two rotational degrees of freedom do not impact the magnetoreceptive properties of the protein. The result implies that any membrane-associated protein is sufficiently restricted in its motion to function as a radical pair-based magnetoreceptor. We relate our theoretical findings to the cryptochromes, currently considered the likeliest candidate to furnish radical pair-based magnetoreception.