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Context. Serendipitous X-ray surveys have proven to be an efficient way to find rare objects, for example tidal disruption events, changing-look active galactic nuclei (AGN), binary quasars, ultraluminous X-ray sources (ULXs), and intermediate mass black holes. With the advent of very large X-ray surveys, an automated classification of X-ray sources becomes increasingly valuable.Aims. This work proposes a revisited naive Bayes classification of the X-ray sources in the Swift-XRT and XMM-Newton catalogs into four classes -- AGN, stars, X-ray binaries (XRBs) and cataclysmic variables (CVs) -- based on their spatial, spectral and timing properties and their multiwavelength counterparts. An outlier measure is used to identify objects of other natures. The classifier is optimized to maximize the classification performance of a chosen class (here XRBs) and it is adapted to data mining purposes.Methods. We augmented the X-ray catalogs with multiwavelength data, source class, and variability properties. We then built a reference sample of about 25000 X-ray sources of known nature. From this sample the distribution of each property was carefully estimated and taken as reference to assign probabilities of belonging to each class. The classification was then performed on the whole catalog, combining the information from each property.Results. Using the algorithm on the Swift reference sample we retrieved 99%, 98%, 92% and 34% of AGN, stars, XRBs, and CVs, respectively, and the false positive rates are 3%, 1%, 9% and 15%. Similar results are obtained on XMM sources. When applied to a carefully selected test sample, representing 55% of the X-ray catalog, the classification gives consistent results in terms of distributions of source properties. A substantial fraction of sources not belonging to any class is efficiently retrieved using the outlier measure, as well as AGN and stars with properties deviating from the bulk of their class. Our algorithm is then compared to a random forest method; the two showed similar performances but the algorithm presented in this paper improved insight into the grounds of each classification.Conclusions. This robust classification method can be tailored to include additional or different source classes and can be applied to other X-ray catalogs. The transparency of the classification compared to other methods makes it a useful tool in the search for homogeneous populations or rare source types, including multi-messenger events. Such a tool will be increasingly valuable with the development of surveys of unprecedented size, such as LSST, SKA and Athena, and the search for counterparts of multi-messenger events., Comment: Astronomy and Astrophysics - A&A, EDP Sciences, inPress