If we read through Chapter 13 of the Brief Second Edition of Traditions & Encounters (the chapter entitled “India and the Indian Ocean Basin”), we can see that there were various attempts to bring the faiths of Islam and Hinduism together. Some were informal while at least one was much more formal.
On the informal level, Bentley and Ziegler say that there were rulers who tried to allow Islam and Hinduism to coexist peacefully. They say (p. 258) that
Ruling elites who converted to Islam often continued to honor Hindu, Buddhist, or native southeast Asian traditions.
The authors argue that these elites wanted to make sure that all of their subjects could coexist peacefully and that they, themselves, could get along with Muslim foreigners. They were trying to bring the various branches of belief together simply because that was convenient for them and for their ability to rule effectively.
A second force for unification was the Sufi movement. The Sufis did not literally try to synthesize Islam and Hinduism. However, they did allow their followers to continue to observe practices from other religions. In this way, they effectively allowed for some sort of synthesis to arise.
By contrast, the bhakti movement did try to pull Hinduism and Islam together. The bhakti movement was originally a Hindu movement, but it came into contact with Islam and was attracted to some Islamic values. It eventually started to explicitly try to connect the two religions. For example, it tried to present Shiva, Vishnu, and Allah as “manifestations of a single, universal deity” (p. 254).
Thus, there were attempts, both formal and informal, to bring the various branches of religious belief in India together.