Is it permissible for a wife to return to her ex-husband without informing him of her previous marriage, knowing that she will be traveling to non-Muslim countries?

You must refer to a scholar to clarify some unclear phrases in the question. Generally, if "married without khalwah" (seclusion) means that her husband did not consummate the marriage with her, nor did he seclude himself with her in a valid khalwah, then if she is divorced, she does not have to observe an iddah (definition) (waiting period), and it is permissible for her to marry someone else. However, if he consummated the marriage with her or secluded himself with her in a valid khalwah, then she must observe the iddah, and marrying her during her iddah is void.

Regarding your statement: "And her guardian was contacted, and he agreed," if the intention is the guardian's permission at the time of the marriage contract, then the marriage is valid. However, if the marriage took place without his permission, and then you sought his approval and he agreed, this marriage is void (fasid (definition)) and separation must occur through annulment (faskh (definition)) or divorce.

As for your statement: "And now she wants to return to her ex-husband without informing him that she married me," it is not permissible for her to return to her first husband until she is free from impediments (such as being under another husband or being in iddah). If the impediments are removed, it is permissible for her to return with a new contract.

Two points should be emphasized: 1. It is not permissible to incite a woman to divorce her husband. Some scholars have even held that the marriage of a woman to someone who incited her against her husband is invalid. 2. Emotional relationships between non-mahram (definition) individuals are forbidden and dangerous.

One must be cautious in matters concerning private parts and inquire about Sharia (definition) rulings before taking action. We reiterate the importance of speaking directly with a scholar.