Strangia is a country mentioned in all three books and possibly the movie, though is never visited. It borders Ingary, High Norland, and possibly the Sultanates of Rashpuht to the south, or at least the desert between.
Tensions between Ingary and Strangia are constantly increasing throughout the course of the story. Fearing that war is imminent, the King of Ingary doubles his search for Prince Justin and Wizard Suliman, and decides to make Howl Jenkins Pendragon a Royal Wizard. However, war does not come until after Wizard Suliman has returned, giving Ingary two royal wizards to serve the army.
Strangia and Ingary do go to war some time after the events of Howl's Moving Castle. The Strangian soldiers are completely overrun by Ingary, due to Ingary's heavy use of magic against Strangia's ordinary army. A peace treaty was signed, and Ingary's Prince Justin was sent to reside there. He was quickly betrothed to Princess Beatrice, the heir to Strangia's throne, to ally the two kingdoms.
Soon after her betrothal, Princess Beatrice was kidnapped by the Djinns, and Prince Justin rushes off to find her.
Strangia may be the "neighboring kingdom" mentioned in the movie that Ingary goes to war with, and the home of the Prince.