  The concept of thermophase refers to the appearance of a phase gradient
inside a superconductor originating from the presence of an applied temperature
bias across it. The resulting supercurrent flow may, in suitable conditions,
fully counterbalance the temperature-bias-induced quasiparticle current
therefore preventing the formation of any voltage drop, i.e., a thermovoltage,
across the superconductor. Yet, the appearance of a thermophase is expected to
occur in Josephson-coupled superconductors as well. Here we theoretically
investigate the thermoelectric response of a thermally-biased Josephson
junction based on a ferromagnetic insulator. In particular, we predict the
occurrence of a very large thermophase which can reach $\pi/2$ across the
contact for suitable temperatures and structure parameters, i.e., the
quasiparticle thermal current can reach the critical current. Such a
thermophase can be several orders of magnitude larger than that predicted to
occur in conventional Josephson tunnel junctions. In order to assess
experimentally the predicted very large thermophase we propose a realistic
setup realizable with state-of-the-art nano fabrication techniques and
well-established materials which is based on a superconducting quantum
interference device. This effect could be of strong relevance in several
low-temperature applications, for example, for revealing tiny temperature
differences generated by coupling the electromagnetic radiation to one of the
superconductors forming the junction.
