  In the fluctuation regime that precedes the onset of pairing in
multi-component superconductors, such as nematic and chiral superconductors,
the normal state is generally unstable towards the formation of charge-$4e$
order $-$ an exotic quantum state in which electrons form coherent quartets
rather than Cooper pairs. However, charge-$4e$ order is often suppressed by
other competing composite orders, such as nematics. Importantly, the formation
of nematic domains is unavoidable due to the long-range strains generated,
leading to one-dimensional regions where the competing nematic order is
suppressed. Here, we employ a real-space variational approach to demonstrate
that, in such nematic domain walls, charge-$4e$ order is locally condensed via
a vestigial-order mechanism. We explore the experimental manifestations of this
effect and discuss materials in which it can be potentially observed.
