The phrase "running across" typically means encountering or happening upon something unexpectedly. There are several synonyms that can be used in place of "running across" to convey the same meaning. One such synonym is stumbling upon, which implies a sense of surprise or serendipity. Another alternative for "running across" is chancing upon, which suggests a random encounter. Similarly, the term happening upon can be used interchangeably with "running across" to convey the idea of finding something unexpectedly. Other synonyms worth considering include encountering, discovering, and stumbling upon. Ultimately, the choice of synonym depends on the context and tone of the sentence or phrase where it is being used.