The $\rho$ Oph region revisited with Gaia EDR3
  Context. Young and embedded stellar populations are important probes of the
star formation process. Paradoxically, we have a better census of nearby
embedded young populations than the slightly more evolved optically visible
young populations. The high accuracy measurements and all-sky coverage of Gaia
data are about to change this situation. Aims. This work aims to construct the
most complete sample to date of YSOs in the $\rho$ Oph region. Methods. We
compile a catalog of 1114 Ophiuchus YSOs from the literature and crossmatch it
with the Gaia EDR3, Gaia-ESO and APOGEE-2 surveys. We apply a multivariate
classification algorithm to this catalog to identify new, co-moving population
candidates. Results. We find 191 new high-fidelity YSO candidates in the Gaia
EDR3 catalog belonging to the $\rho$ Oph region. The new sources appear to be
mainly Class III M-stars and substellar objects and are less extincted than the
known members. We find 28 previously unknown sources with disks. The analysis
of the proper motion distribution of the entire sample reveals a well-defined
bimodality, implying two distinct populations sharing a similar 3D volume. The
first population comprises young stars' clusters around the $\rho$ Ophiuchi
star and the main Ophiuchus clouds (L1688, L1689, L1709). In contrast, the
second population is older ($\sim$ 10 Myr), dispersed, has a distinct proper
motion, and is possibly from the Upper Sco group. The two populations are
moving away from each other at about 4.1 km/s, and will no longer overlap in
about 4 Myr. Finally, we flag 17 sources in the literature as impostors, which
are sources that exhibit large deviations from the average distance and proper
motion properties of the $\rho$ Oph population. Our results show the importance
of accurate 3D space and motion information for improved stellar population
analysis. (Abridged)
