  We release the AllWISE counterparts and Gaia matches to 106,573 and 17,665
X-ray sources detected in the ROSAT 2RXS and XMMSL2 surveys with |b|>15. These
are the brightest X-ray sources in the sky, but their position uncertainties
and the sparse multi-wavelength coverage until now rendered the identification
of their counterparts a demanding task with uncertain results. New all-sky
multi-wavelength surveys of sufficient depth, like AllWISE and Gaia, and a new
Bayesian statistics based algorithm, NWAY, allow us, for the first time, to
provide reliable counterpart associations. NWAY extends previous distance and
sky density based association methods and, using one or more priors (e.g.,
colors, magnitudes), weights the probability that sources from two or more
catalogues are simultaneously associated on the basis of their observable
characteristics. Here, counterparts have been determined using a WISE
color-magnitude prior. A reference sample of 4524 XMM/Chandra and Swift X-ray
sources demonstrates a reliability of ~ 94.7% (2RXS) and 97.4% (XMMSL2).
Combining our results with Chandra-COSMOS data, we propose a new separation
between stars and AGN in the X-ray/WISE flux-magnitude plane, valid over six
orders of magnitude. We also release the NWAY code and its user manual. NWAY
was extensively tested with XMM-COSMOS data. Using two different sets of
priors, we find an agreement of 96% and 99% with published Likelihood Ratio
methods. Our results were achieved faster and without any follow-up visual
inspection. With the advent of deep and wide area surveys in X-rays (e.g.
SRG/eROSITA, Athena/WFI) and radio (ASKAP/EMU, LOFAR, APERTIF, etc.) NWAY will
provide a powerful and reliable counterpart identification tool.
