The phenomenon of multiple institutional affiliations of authors of research papers, or synchronous scientific mobility has emerged, partly in response to a scientometric quantitative approach to assessing the performance of scientific organizations in full accordance with Goodhart's law. Its wide distribution may distort the results of analysis of research collaborations based on bibliometric data. The article traces the influence of this phenomenon on the assessment of international collaborations of Russian researchers and organizations, evaluates the success of Russian programmes to attract leading foreign scientists. We show that in up to 20% of Russian international publications there are no authors with purely Russian affiliations. We identified at least 225 presumably invited researchers who have not published a single paper in collaboration with their Russian colleagues in 2014-2018.