नाथमुनिः
पूर्वाचार्य of the विशिष्ताद्वैत-श्रीवैष्णव सम्प्रदाय; he who single-handedly retrieved the दिव्यप्रबन्ध, an exquisite tamizh canon of 4000 verses for विष्णु, from obscurity even as वराहदेव retrieves the earth from the waters.
Date
विद्यानद्यम्बुराशिं विमल-तर-गुण-व्रात-रत्नाकरो यः
पाराशर्यान्ववायाम्बुधिशिशिरकरो भट्टदत्ताभिधानः,
तस्य श्रीनाथपादाम्बुरुहयुगळिकाषट्पदस्य प्रशस्तिम्
पुत्रश्चित्राद्धिरम्यां समरचयद् इमाम् माधवो भट्टयज्वा ॥
anbil plates issued in 962
So, shrInAtha / nAthamuni can be dated to late 9th century CE.
What we do know is that alavandar was too old by the time ramanuja meets him.. even if we ignore the dates given for alavandar (yamuna) and nathamuni, this much is clear.. yamuna must have been born before 950ce at the latest.. most probably by 940 or before.. in that case, again nathamunis birth cannot be placed later than 900.