Is a woman considered Islamically divorced after separating from her husband for 10 months without a formal divorce or communication? And is it permissible Islamically for a man and a woman to marry solely by mutual consent, without witnesses, paperwork, or a contract?

The physical separation between spouses does not signify the dissolution of the marital bond. The marriage contract is only dissolved by divorce, annulment, or death.

Therefore, the sister's separation from her husband for the aforementioned period is not considered a divorce.

As for the mentioned marriage, it is an unlawful and invalid marriage due to the absence of the necessary Islamic legal conditions for marriage.