Yonimo Osoroshii Nihon Mukashibanashi

Both are collection of short horror stories of Japanese origin. While Yami Shibai has four minute long stories, Yonimo Osoroshii Nihon Mukashibanashi has 18 minute long stories.  

They're both early 2000s horror anime, that make a honest attempt at disturbing the viewer. Both share that grim underlying feeling absent from modern anime. It's a safe bet. 

Both are collection of short horror stories. While Yami Shibai has 13 four minute long stories, Yonimo Osoroshii Nihon Mukashibanashi has three 18 minute long stories. 

Both are OVAs containing three short stories which takes common stories (Grimm fairy tales/Japanese folklore stories) and remake them into something frightening. Both are dementia and horror genre. 

Both take look at otherwise fairly well know stories (Village Doctor by Franz Kafka/traditional japanese fairy tales) and remake them into something weird and creepy. Both are dementia and psychological genre.  

Both are collection of stories from japanese folklore. While Fuy no Hi is made more artistic (different styles of animation for each part), Yonimo Osoroshii Nihon Mukashibanashi is made more as a horror/dementia.