I see, a partial ordering is not enough to reliably search for keys. Imagine doing a binary search on a set with the trivial partial ordering (nothing compares to anything).
[ A, B, C, D, E, F, G ]
To find A: check is A <= D (no), so restrict the search to [ E, F, G ] and you can see that we’ve already failed.
So in the case of keying it makes sense to have an “arbitrary total ordering” just for the sake of having a total ordering; on the other hand, the exact ordering will not make sense for objects that don’t have a natural total ordering. This is an unfortunate state of affairs and it’s not clear to me what should be done about it.