Serbians always ask for your passport when you check into a hotel or any apartment. If you don't have it, they start to really worry and don't want to let you stay. I had to turn in my passport at the embassy, but I needed to spend the night somewhere other than home. I had my driver's license on me and a photo of my passport, so I figured we could work it out.
What made things complicated was the border with Bosnia right at the edge of the city. The city literally ended at the checkpoint. Everyone's paranoia level was cranked to maximum. They did let me check in, but I practically had to recite my entire family tree. And they told me my driver's license wasn't a valid document :)
The hotel owner strongly urged me not to wander around the city in the evening. According to him, there's a lot of police on the streets, and they're suspicious of outsiders. Without a passport, they could take you to the station for an identity check. If it comes out that they let someone without a passport stay, the hotel could face problems too. Then he went on at length about crowds of Arabs and Iranians trying to illegally cross into Bosnia and then into the EU. If the internet is to be believed, Serbian police really aren't very friendly toward illegal immigrants. There are even unpleasant incidents (if you don't want to see someone getting beaten, don't click it).
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