As we have seen, according to Rambam, Rosh, and Ran the principle of ein bishul aĥar bishul applies to liquids as well as solids. Thus as long as the liquids were fully cooked, even if they have now cooled off, one may reheat them to yad soledet bo.
There are many Yemenite Jews who consistently follow Rambam’s rulings throughout halakha. Therefore, if they serve soup Friday night and then put it in the refrigerator, they may remove the soup from the refrigerator on Shabbat day and reheat it. This is the law for sauces and all other liquids as well.
Ashkenazim follow Rema, whose opinion is a kind of compromise between the previous two. If a liquid has cooled off to the point that one normally would therefore not consume it, then one may not reheat it to yad soledet bo. But if it is still warm, one may reheat it to yad soledet bo. This is because Rema fundamentally accepts the position of Rambam that ein bishul aĥar bishul applies to liquids as well. However, in his opinion, if a liquid has cooled down entirely, it is rabbinically forbidden to reheat it.
When one who follows the position of Shulĥan Arukh or Rema is a guest at the home of a Yemenite who follows Rambam, he may eat soup that his host heated on Shabbat. Since the host is following halakha based on the accepted practice of his community, any Jew may eat his food le-khatĥila (see MB 318:2.).
However, one who follows the Shulĥan Arukh or Rema may not ask one who follows Rambam to heat soup for him. Since his own community forbids reheating soup, he may not ask his friend to do it for him. But if he invites his friend to eat with him, his friend may follow his own custom and heat soup for himself. In such a case, the host may eat from it as well.