Is what we did, getting married without a guardian (wali) or witnesses in a non-Muslim country, considered adultery (zina), knowing that we later documented the marriage after returning to the United States?

Marriage in Islam has pillars and conditions, which are: proposal and acceptance (ijab and qabul), designation of the spouses, their consent, the presence of a guardian (wali), the testimony of two just Muslim witnesses, and publicizing the marriage (ishhar). If a woman does not have a Muslim guardian, a Muslim judge (qadi) should marry her off. If there is no Muslim judge, then a prominent Muslim man should marry her off. The civil marriage conducted by American courts is not considered a valid marriage according to Islamic law, due to the absence of some of its pillars and conditions. Rather, it is sexual intercourse under a semblance of validity (wat' bi shubhah), which entails the legal effects of a marriage contract. However, the contract must be re-executed within an Islamic framework, fulfilling all its Islamic pillars and conditions.