Wit Studio (Attack on Titan, Kabaneri of the Iron Fortress, The Ancient Magus' Bride) is handling the animation. The anime is slated to premiere in Japanese theaters this winter.
Though Shochiku's page focuses on the Laughing Under the Clouds Gaiden anime, it also includes an image of the upcoming live-action Laughing Under the Clouds film's cast. The film will premiere in February 2018. Katsuyuki Motohiro (Psycho-Pass chief director) is directing the film, and Yuuya Takahashi (Laughing Under the Clouds anime series, Tiger & Bunny, Lupin the Third) wrote the script.
Karakara launched Donten ni Warau Gaiden on Mag Garden's "Beat's" manga site in 2014 after running the original Donten ni Warau manga in Mag Garden's Monthly Comic Avarus magazine from 2011-2013. It ended on February 3.