  More than 50% of solar-mass stars form in multiple systems. It is therefore
crucial to investigate how multiplicity affects the star and planet formation
processes at the protostellar stage. We report continuum and C$^{18}$O (2-1)
observations of the VLA 1623-2417 protostellar system at 50 au angular
resolution as part of the ALMA Large Program FAUST. The 1.3 mm continuum probes
the disks of VLA 1623A, B, and W, and the circumbinary disk of the A1+A2
binary. The C$^{18}$O emission reveals, for the first time, the gas in the
disk-envelope of VLA 1623W. We estimate the dynamical mass of VLA 1623W,
$M_{\rm dyn}=0.45\pm0.08$ M$_{\odot}$, and the mass of its disk, $M_{\rm
disk}\sim6\times10^{-3}$ M$_{\odot}$. C$^{18}$O also reveals streamers that
extend up to 1000 au, spatially and kinematically connecting the envelope and
outflow cavities of the A1+A2+B system with the disk of VLA 1623W. The presence
of the streamers, as well as the spatial ($\sim$1300 au) and velocity
($\sim$2.2 km/s) offset of VLA 1623W suggest that either sources W and A+B
formed in different cores, interacting between them, or that source W has been
ejected from the VLA 1623 multiple system during its formation. In the latter
case, the streamers may funnel material from the envelope and cavities of VLA
1623AB onto VLA 1623W, thus concurring to set its final mass and chemical
content.
