A Jewish couple gets a civil divorce. The wife remarries, and has children with her new spouse. Are the children of this second marriage, which was performed without a get having been issue, mamzerim? Is the wife guilty of adultery?
It doesn't sound like any semblance of the former (a document handwritten by a Jew for the divorce of this specific couple including their names, the date, specific formulations, signed kosher witnesses, etc. presented to the wife by the husband again in front of kosher witnesses, etc.) has taken place. There's very little wiggle room with it and it is usually arranged for by trained professionals.
So they are still married. Any extramarital relations would be adultery and any progeny thereof would be mamzerim. Not a good situation.
(If it turns out that the original marriage wasn't kosher, then problem solved. But that has to be evaluated on a case-by-case basis.)