We prove a satellite formula for the real Floer homotopy types of knots with odd satellite patterns, using an excision cobordism argument in real Seiberg–Witten theory.
Unlike the case of Heegaard Floer knot homology, the real Floer homotopy types do not sensitively depend on the choice of satellite pattern.
We also show that the local equivalence class of the real Floer homotopy type of a knot depends only on the diffeomorphism type of its 0-surgery.
These results yield several applications.