In the beginning, there was The Mistake, and from that The Snowball ...
The Mistake
When the original ceasefire was negotiated 20 years prior, this piece of land where this city (or town) was located was accidentally not portioned out to either side. As it was, at the time, of low significance and value it was not something that either side thought about or sought to claim. Because nobody claimed this land in the ceasefire, it ended up in a sort of limbo -- part of the country, but at the same time, not under the control of either government.
At the time, it was deemed a minor mistake, and not one worth bringing up in the event that it would jeopardize the entire ceasefire treaty. In effect, the politicians that noticed the error in the treaty decided to sacrifice these people to ensure that the treaty was finalized and adopted.
The Rolling Snowball
Much like a little snowball that gets bigger as one rolls it along the ground, what was a small issue grew over the years.
The now neutral area gained value specifically because it was not under either governments control by the very treaties that made them legitimate, but at the same time they are still part of the country. Businessman and criminal alike flock to the area to take advantage of the neutral turf for their on reasons.
A municipal government that was once desperately trying to hold their city together after the war plagued by an influx of people and increased need for supplies. While there was some shorter term strife, now 20 years later this municipal government is stronger and able to hold their own -- even if for a short time
As the unclaimed city started to thrive, it opened up a new opportunity for both sides to have a neutral place to trade and to communicate without encroaching on each other's territory. This meant that previously unavailable or hard to get goods suddenly had a place that they could be acquired. As they saw value in a neutral territory, both sides invested into it to make sure that it remained out of the other's hands.
Inertia
Now as to the question of "What plausible political reasons could they have for allowing it to continue to exist?", there are a couple of answers.
The first is above -- both sides now have a use for a neutral city. And so long as is is more valuable as neutral ground then their ground it will remain neutral. At some point inertia sets in, in that this city has been neutral for a generation so it should stay that way.
The second reason is more political and simpler to explain. A mistake was made in the treaty and after this long, nobody wants to admit that one was made. To admit that they were wrong would bring peril to their political position for reasons, which could possibly inconveniently weaken their position in such a delicate time.