What script was invented by Kublai Khan's imperial preceptor?

Kublai Khan’s imperial preceptor, Drogön Chögyal Phagpa, invented the Phags-pa script. He designed it in the 13th century at Kublai’s request as a unified writing system for the multilingual Mongol Empire.

Phagpa was a Tibetan monk of the Sakya school who became a powerful religious adviser at the Yuan court. Around 1269, he adapted features of the Tibetan alphabet to represent languages including Mongolian and Chinese. The resulting script was also known as the “square script” because of its distinctive geometric appearance.

Kublai Khan intended Phags-pa to function across his empire, helping the administration communicate in a realm that used several writing traditions. It was proclaimed an official script, and examples survive on seals, inscriptions, coins, monuments, and official documents.

The script never displaced the older Uyghur-derived Mongolian script in everyday use. That is the main source of confusion in this question: Phags-pa was based on Tibetan, but it was neither simply the Tibetan script nor the traditional Mongolian alphabet. Its creator was later formally titled Imperial Preceptor.