The phrase "set house together" can be replaced with numerous synonyms, depending on the context. For example, one could use "move in together," "establish a household," "reside together," or simply "live together." Other alternatives include "cohabit," "share a home," "build a life together," "combine households," or even "pool resources and create a living arrangement." These synonyms can be used interchangeably in most situations, and each one conveys a slightly different nuance. Ultimately, the choice of synonym depends on the speaker's tone and intention, as well as the audience's familiarity with the particular idiom.