1. Note that we do the marriage ceremony at least three ways, any of which in theory would be enough. It would help if there wasn’t any hupah, kedushin, or biah as well, and no one knew they were living together as man and wife.
2. Arguing that the eidim are treff is an idea, depending on how desperate you are? It would certainly help if the married couple were clearly not frum (never went to mikvah, never kept kosher, never were Shomer Shabbos – thought the ceremony was something to amuse the grandparents and all that mattered was a civil ceremony).
3. Remember the only big change of a marriage being annuled under halacha is that the woman doesn’t need a “get”. Unlike the traditional goyish law in western countries, children of an unmarried Jewish couple are kosher (not “bastards”) and inherit from the father.
4. How desperate is it to treff up the marriage? Presumably the likely problem is that the woman ended up having children by a different man, and we don’t want those children to be mamzerim. In these situation, our rabbanim are quite creative in finding a basis to decide the marriage was treff. On the other hand, if the dispute is whether the woman gets a kesubah when the husband runs off with a trophy wife (after her being a good housewife for 20 years), the husband’s arugment that the witnesses were treff (and he doesn’t owe his long suffering virtuous wife any money) might not be as well received. If you look at actual cases (tseuvahs) rabbanim are really good lawyers and always have been (even if they will insist they are nothing like lawyers, which is really false modesty).