  We introduce two notions of coarse embeddability between operator spaces:
almost complete coarse embeddability of bounded subsets and
spherically-complete coarse embeddability. We provide examples showing that
these notions are strictly weaker than complete isomorphic embeddability - in
fact, they do not even imply isomorphic embeddability. On the other hand, we
show that, despite their nonlinearity, the existence of such embeddings
provides restrictions on the linear operator space structures of the spaces.
Examples of nonlinear equivalences between operator spaces are also provided.
