  We construct a geometric lifting of the Burge correspondence as a composition
of local birational maps on generic Young-diagram-shaped arrays. We establish
its fundamental relation to the geometric Robinson-Schensted-Knuth
correspondence and to the geometric Sch\"utzenberger involution. We also show a
number of properties of the geometric Burge correspondence, specializing them
to the case of symmetric input arrays. In particular, our construction shows
that such a mapping is volume preserving in log-log variables. As an
application, we consider a model of two polymer paths of given length
constrained to have the same endpoint, known as polymer replica. We prove that
the distribution of the polymer replica partition function in a log-gamma
random environment is a Whittaker measure, and deduce the corresponding
Whittaker integral identity. For a certain choice of the parameters, we notice
a distributional identity between our model and the symmetric log-gamma polymer
studied by O'Connell, Sepp\"al\"ainen, and Zygouras (2014).
