The Shiva Purana counts Hanuman among Shiva's own avatars, and popular Shaiva tellings expand it: seeing Vishnu about to descend as Rama, Shiva chose to descend as well, as the monkey Hanuman, the eleventh Rudra, so that he could serve his friend in person. Texts named for the idea include the Mahabhagavata, Skanda and Brihaddharma Puranas and the Mahanataka; a folk tale has Shiva visit Ravana in this shape unrecognised. Pattanaik summarises the Puranic range — son, manifestation, avatar or eleventh Rudra of Shiva — and that summary is his.
Base telling for this frame: Hanuman's devotion to Rama is Shiva's devotion to Vishnu.
Hanuman — Wikipedia; 11 Rudra avatars of Mahadeva — Exotic India (article); Hanuman, the Mighty Hero — Devdutt Pattanaik
Vaishnava schools hold Hanuman to be the son and incarnation of the wind god Vayu, born to Anjana, and read his service to Rama as the perfect devotee's service to Vishnu without any Shaiva descent. South Indian texts offer a third reading, Hanuman as a union of Shiva and Vishnu, linked to the Ayyappa idea, and one seventeenth-century source has all three gods combine to become him.
No Shiva-avatar: parentage from Vayu; the combined-form readings are further alternatives.
Sources: Hanuman — Wikipedia; 11 Rudra avatars of Mahadeva — Exotic India (article); Hanuman, the Mighty Hero — Devdutt Pattanaik
Thailand's Ramakien makes Phra Ram an incarnation of Phra Narai and a Bodhisatta; Cambodia's Reamker reads Preah Ream almost as a mortal through a Theravada lens.
Gods and demons churn the milk ocean with Mount Mandara on the tortoise's back; poison, treasures, Lakshmi and the nectar emerge, and Vishnu as Mohini keeps the nectar for the gods.
Rama worships Shiva as a linga on the island of Rameswaram — before the bridge in the Ramayana, or after Ravana's death to expiate the killing of a brahmin in temple tradition, where Sita's sand linga and Hanuman's linga from Kailasa stand side by side.
Rama wishes to worship Shiva on the island before crossing to Lanka, or after killing Ravana, to be cleansed of killing a brahmin.
Vishnu offers a thousand lotuses and his own eye at Kashi; Rama raises a sand linga at Rameswaram before crossing to Lanka; the full record is I-10 in the interactions file.
Rama's return in the north, Narakasura's death in the south and west, Lakshmi's wedding night in Puranic lore, and Bali's return in the Deccan and Gujarat.
Shiva and Parvati must choose between their two sons.
The Maitri Upanishad names Brahma, Rudra and Vishnu together as portions of one being, matched to the three gunas.