"Keep shtum" is an informal idiom that means to stay quiet or to keep silent. It is often used to indicate that someone should not reveal or disclose sensitive or confidential information. There are several synonyms for this phrase that you can use if you want to avoid using slang or informal language. Some examples include "stay mum," "keep quiet," "remain silent," "hold your tongue," or "say nothing." These phrases can be used interchangeably with "keep shtum" in different contexts, such as in business, politics, or personal relationships. Ultimately, it's up to you to choose the most appropriate synonym for your particular situation.