Question: Are Tungabhadra (Kudali) Sringeri and Rameswaram (Rama-Kshetram) two different locations or Are they referring to the same place?

Answer:

“He (Sri Sankaracharya), retired to the banks of Tungabhadra and fixed His camp at Rameswara Tirtha.” (A History of South India-Sri K. A. N. Sastri)

This would mean that Tungabhadra Sringeri and Rameswara kshetra are identical places.