If you're looking for anime similar to Patlabor: The Mobile Police, you might like these titles. All recommendations are made by Anime-Planet users like you!
The government has passed the Media Betterment Act, establishing a military police force to stop the spread of "negative influences harmful to society;" and in response, libraries have organized their own military units to protect freedom of expression. After being saved by a Library Defense Force agent in a bookstore, Kasahara is inspired to enlist in the Library Defense Force. Although she proves quite athletic and capable, she must endure the harsh but meaningful training of her instructor and superior officer, Doujou. As she learns how to be a good soldier, she participates in LDF operations, helping protect literary freedom for everyone.
Both are slightly futuristic (Patlabor more so), and focuses about new and special branches of government: library military forces and police mecha task forces. There is lots of focus on characters, a good amount of humour and are very down to earth.
If you liked either one you should do yourself a favour and check the other one out.