  The present paper is concerned with the well-posedness theory for
non-homogeneous incompressible fluids exhibiting odd (non-dissipative)
viscosity effects. Differently from previous works, we consider here the full
odd viscosity tensor. Similarly to the work of Bresch and Desjardins in
compressible fluid mechanics, we identify the presence of an effective velocity
in the system, linking the velocity field of the fluid and the gradient of a
suitable function of the density. By use of this effective velocity, we propose
a new formulation of the original system of equations, thus highlighting a
strong similarity with the equations of the ideal magnetohydrodynamics. By
taking advantage of the new formulation of the equations, we establish a local
in time well-posedness theory in Besov spaces based on $L^\infty$ and prove a
lower bound for the lifespan of the solutions implying ``asymptotically
global'' existence: in the regime of small initial density variations,
$\rho_0-1= O(\varepsilon)$ for small $\varepsilon>0$, the corresponding
solution is defined up to some time $T_\varepsilon>0$ satisfying the property
$T_\varepsilon\,\longrightarrow\,+\infty$ when $\varepsilon\to0^+$.
