  In this work we propose a new gravitational setup formulated in terms of two
interacting vierbein fields. The theory is the fully diffeomorphism and local
Lorentz invariant extension of a previous construction which involved a fixed
reference vierbein. Certain vierbein components can be shifted by local Lorentz
transformations and do not enter the associated metric tensors. We parameterize
these components by an antisymmetric tensor field and give them a kinetic term
in the action, thereby promoting them to dynamical variables. In addition, the
action contains two Einstein-Hilbert terms and an interaction potential whose
form is inspired by ghost-free massive gravity and bimetric theory. The
resulting theory describes the interactions of a massless spin-2, a massive
spin-2 and an antisymmetric tensor field. It can be generalized to the case of
multiple massive spin-2 fields and multiple antisymmetric tensor fields. The
absence of additional and potentially pathological degrees of freedom is
verified in an ADM analysis. However, the antisymmetric tensor fluctuation
around the maximally symmetric background solution has a tachyonic mass pole.
