  We investigate linear operators between C$^\ast$-algebras which approximately
preserve involution and orthogonality, the latter meaning that for some
$\varepsilon>0$ we have $\|\phi(x)\phi(y)\|\leq\varepsilon\|x\|\|y\|$ for all
positive $x,y$ with $xy=0$. We establish some structural properties of such
maps concerning approximate Jordan-like equations and almost commutation
relations. In some situations (e.g. when the codomain is finite-dimensional),
we show that $\phi$ can be approximated by an approximate Jordan
$^\ast$-homomorphism, with both errors depending only on $\|\phi\|$ and
$\varepsilon$.
