There's no apparent relation between the Soredemo Machi wa Mawatteiru and Natsu no Arashi manga series—they were published in different magazines and have different authors (Natsu no Arashi is by Jin Kobayashi, the author of School Rumble). From what I can tell, the stories are completely different too. (I've only seen one episode of Soredemo Machi wa Mawatteiru, and never anything of Natsu no Arashi.) But the anime versions of both shows are produced by Shaft and directed by Akiyuki Shinbou, as their Wikipedia pages point out. That's probably what the TVTropes page was referring to.
I'm not really sure how that makes Soredemo Machi wa Mawatteiru a spiritual successor to Natsu no Arashi—look at Shinbou's filmography and you could claim that, say, The Soul Taker was the spiritual successor to Twilight of the Dark Master, or Denpa Onna was the spiritual successor to Arakawa Under the Bridge, on the basis that Shinbou directed one and then later directed the other. On the other hand, the otherwise unrelated Pani Poni Dash and Negima?! do share a lot of stylistic similarities, so perhaps it's something like that.