  Hurwitz numbers enumerate branched morphisms between Riemann surfaces. For a
fixed elliptic target, Hurwitz numbers are intimately related to mirror
symmetry following work of Dijkgraaf. In recent work of Chapuy and Dolega a new
variant of Hurwitz numbers with fixed genus $0$ target was introduced that
includes maps between between non-orientiable surfaces. These numbers are
called $b$-Hurwitz numbers and are polynomials in a parameter $b$ which
measures the non-orientability of the involved maps. An interpretation in terms
of factorisations of $b$-Hurwitz numbers for $b=1$, so-called twisted Hurwitz
numbers, was found in work of Burman and Fesler. In previous work, the authors
derived a tropical geometry interpretation of these numbers. In this paper, we
introduce a natural generalisation of twisted Hurwitz numbers with elliptic
targets within the framework of symmetric groups. We derive a tropical
interpretation of these invariants, relate them to Feynman integrals and derive
an expression as a matrix element of an operator in the bosonic Fock space.
