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In order to accurately answer this question, we need more information, in particular: for how long is a U.S. citizen on a tourist visa or no visa going to stay in Germany?
According to Reporting presence for short stays:
If you are moving into an apartment or a house, for up to 3 months, there is no obligation to notify the registration authority.
If you stay for more than 3 months, didn't register your local address within the predefined deadline, get caught, and if there are no particular reasons preventing you from registering within the statutory time limit, you should pay a fine that can be up to β¬ 500
I have never met a landlord that required to show proof of registration. However, this is just my personal experience.