Would the laws of taharat mishpacha kick in, for example.
They should definitely keep taharat hamishpacha, which is irrelevant of what exact marriage ceremony they did/didn't have.
We would advise such a couple to go through a proper Jewish ceremony when they can; on the other hand, there is no halachic stigma on children born from two Jewish parents who weren't halachically wed.
But for theory's sake, this question was debated hotly in twentieth-century America: Rabbi Yosef Eliyahu Henkin z"l felt that as it's public knowledge that this couple is having marital relations, that constitutes Jewish marriage (albeit not the recommended route for it!). (The Gemara discusses a case where a couple may have in mind that if their ring ceremony was somehow flawed, marital relations would work as a backup.)
Rabbi Moshe Feinstein z"l felt that if necessary, we could argue that the couple chose to opt out of halachic marriage and thus never intended for it to apply to them. Both rabbis agreed that to dissolve such a marriage, a Jewish divorce (Get) is required; Rabbi Feinstein felt that if a Get was unobtainable (or if after the fact, she went out and married someone else and bore a child from them), we could follow the non-marriage argument. This was a bold statement of Rabbi Feinstein (it even got surprised looks from his less-traditionalist cousin-once-removed, Rabbi Joseph Dov Soloveichik), and most rabbis today try not to rely on it.
Rabbi Yissochar Frand describes how Rabbi Y.E. Henkin suffered from Alzheimer's at the end of his life; many of the patients in that hospital unit had to be gagged because of the horrible things coming out of their mouths. Rabbi Henkin, stripped of all his faculties, was heard chanting:
Can't make kiddush before tekios. She needs a Get. Can't make kiddush before tekios. She needs a Get.