Near the end of posting my question, I became convinced that the title might be "X" with some sort of suffix, so I plugged
anime X into Google, and it brought up Clamp's X/1999, which seems like it matches fairly well.
In 1999, a teenager named Kamui Shirō returns to Tokyo after a six-year absence. He comes to protect those dearest to him, Kotori and Fūma Monou, and fulfill his mother's dying wish of changing fate. The end of the world is fast approaching as superhuman individuals gather and take sides in Tokyo, the battle site of Armageddon. Following Kamui's arrival the Dragons of Heaven and the Dragons of Earth, the two factions in the final battle for humanity's future, vie for the young man's allegiance, convinced his power will assure their victory. The Dragons of Heaven are the first to contact Kamui. The Dragons of Heaven are guided by Hinoto, dreamgazer for the Legislature of Japan. They are the protectors of the kekkai (結界), spiritual barriers (in the form of buildings such as the Tokyo Tower) that hold the fabric of nature together. As long as the kekkai survive, Judgment Day is postponed. The Dragons of Earth are the counterpart to the Dragons of Heaven. Their mission is to destroy the kekkai and unleash earthquakes so the Earth can be cured of the plague of humanity. They were assembled by Hinoto's sister Kanoe, secretary to the Governor of Tokyo.
The kekkai do have the relevant effect of protecting the world if the one who puts it up survives the battle.
As the barrier fields created by the Dragons of Heaven, the Spirit Shields are fields of varying geometrical shape (depending on which of the Dragons of Heaven has created the Spirit Shield) which only those involved with the end of the world can exist within. The barrier field effectively creates an alternate plane or dimension resembling the area of the material plane which the Spirit Shield was created to protect. This ensures that (so long as the one who created the Spirit Shield remains alive, conscious, in relatively decent health, and maintains an undisclosed amount of concentration required to create the Spirit Shield in the first place) the area in which the Spirit Shield was created and all the ordinary people within it cannot be harmed. Also, as a general rule, it is impossible for even the Dragons of Heaven and Dragons of Earth themselves to leave the space of a Spirit Shield until the creator of the Spirit Shield releases it. Trying to leave a Spirit Shield is said to be extremely painful and dangerous, requiring a great deal of power to even attempt. The rules regarding entering a Spirit Shield, however, seem to be more obscure.
Another way to easily put it is that a Spirit Shield is like a carbon copy of the battlefield combatants can do battle without fear of hurting innocent people or any interference from the outside. If any damage at all occurs in the city inside the Spirit Shield and the one who puts it up lives or kills the other, then no damage will be done to the real location in the primary dimension—as if the battle had never happened. However, if the one who set up the Spirit Shield in the first place gets injured to the point that he or she cannot maintain the requisite concentration to support the Spirit Shield, falls unconscious, or killed altogether, then the Spirit Shield is destroyed, and the damage done to the location within the Spirit Shield during combat is done to the primary physical dimension, killing many innocent bystanders in the vicinity and causing massive destruction.
The film, released in 1996, has as one of the earliest battles a male in white religious robes fighting a dark-clothed male, and indeed, when the guy in dark clothes wins, the area is shown being destroyed as the kekkai falls.