  Bi-Hamiltonian structures are of great importance in the theory of integrable
Hamiltonian systems. The notion of compatibility of symplectic structures is a
key aspect of bi-Hamiltonian systems. Because of this, a few different notions
of compatibility have been introduced. In this paper we show that, under some
additional assumptions, compatibility in the sense of Magri implies a notion of
compatibility due to Fass\`o and Ratiu, that we dub bi-affine compatibility. We
present two proofs of this fact. The first one uses the uniqueness of the
connection parallelizing all the Hamiltonian vector fields tangent to the
leaves of a Lagrangian foliation. The second proof uses Darboux-Nijenhuis
coordinates and symplectic connections.
