Temporary residence Card

In many cases the reluctance to register the rental agreement for the property with the local police is an attempt for the owner to avoid paying tax on the rental. In other cases it is sometimes the fact that the landlord is not the owner and has actually no right to rent the property.

The rule is that the landlord should take a foreigner's passport along with the House/room/apartment registration document (red book- the title deed) to register the stay of the former in the same ward police station.
As jpbsimpson posted, many 'landlords' are not landlords but just rent the whole house and re rent or the real landlord hesitate to do that because they are renting it illegally (tax evasion). Some will do it even though they are not legal landlord by bribe the police!
FYI : Landlord is not the authority to issue a TRC but once you register in the ward police they can attest your document that you are a resident of that jurisdiction with no criminal background. That document can use in various government departments such as to apply for a TRC/WP/PRC or VEC or to marry a Vietnamese citizen.

And the other reason, and very often this is the case here, they do not have the house book becuase they have borrowed money on the property and the bank is holding the house book. Me, if they do not want to go through a notary I do not consider renting there. To may pitfalls and to much underhanded dealings. Protect yourself, they have no desire to protect you.