  Context: The Carina Nebula complex (CNC) is one of the most massive and
active star-forming regions in our Galaxy and it contains several large young
star clusters. The distances of the individual clusters and their physical
connection were poorly known up to now, with strongly discrepant results
reported in the literature. Aims: We want to determine reliable distances of
the young stellar clusters in the central Carina Nebula region (in particular,
Tr 14, 15, and 16) and the prominent clusters NGC 3324 and NGC 3293 in the
northwestern periphery of the CNC. Methods: We analyzed the parallaxes in Gaia
EDR3 for a comprehensive sample of 237 spectroscopically identified OB stars,
as well as for 9562 X-ray-selected young stars throughout the complex. We also
performed an astrometric analysis to identify members of the young cluster vdBH
99, which is located in the foreground of the northwestern part of the Carina
Nebula. Results: We find that the distances of the investigated clusters in the
CNC are equal within <= 2%, and yield very consistent most likely mean distance
values of 2.36(+0.05 -0.05) kpc for the OB star sample and 2.34(+0.05 -0.06)
kpc for the sample of X-ray-selected young stars. Conclusions: Our results show
that the clusters in the CNC constitute a coherent star-forming region, in
particular with regard to NGC 3324 and NGC 3293 at the northwestern periphery,
which are (within <= 2%) at the same distance as the central Carina Nebula. For
the foreground cluster vdBH 99, we find a mean distance of 441(+2 -2) pc and an
age of ~ 60 Myr. We quantified the contamination of X-ray-selected samples of
Carina Nebula stars based on members of this foreground cluster.
