  Schur coupling (SC) and equivalence after extension (EAE) are important
relations for bounded operators on Banach spaces. It has been known for 30
years that the former implies the latter, but only recently Ter Horst,
Messerschmidt, Ran and Roelands disproved the converse by constructing a pair
of Fredholm operators which are EAE, but not SC.
  Motivated by this result, we investigate when EAE and SC coincide for
Fred\-holm operators. Fredholm operators which are EAE have the same Fredholm
index. Surprisingly, we find that for each integer $k$ and every pair of Banach
spaces $(\mathcal{X},\mathcal{Y})$, either no pair of Fredholm operators of
index~$k$ acting on $\mathcal{X}$ and $\mathcal{Y}$, respectively, is SC, or
every pair of this kind which is EAE is also SC. Consequently, the question
whether EAE and SC coincide for Fredholm operators of index~$k$ depends only on
the geometry of the underlying Banach spaces $\mathcal{X}$ and $\mathcal{Y}$,
not on the properties of the operators themselves.
  We quantify this finding by introducing two numerical indices which capture
the coincidence of EAE and SC, and provide a number of examples illustrating
the possible values of these indices. Notably, this includes an example showing
that the above-mentioned result of Ter Horst et al, which is based on a pair of
essentially incomparable Banach spaces, does not extend to projectively
incomparable Banach spaces.
