  Several fluid systems are characterised by time reversal and parity breaking.
Examples of such phenomena arise both in quantum and classical hydrodynamics.
In these situations, the viscosity tensor, often dubbed ``odd viscosity'',
becomes non-dissipative. At the mathematical level, this fact translates into a
loss of derivatives at the level of \textsl{a priori} estimates: while the odd
viscosity term depends on derivatives of the velocity field, no parabolic
smoothing effect can be expected.
  In the present paper, we establish a well-posedness theory in Sobolev spaces
for a system of incompressible non-homogeneous fluids with odd viscosity. The
crucial point of the analysis is the introduction of a set of \emph{good
unknowns}, which allow for the emerging of a hidden hyperbolic structure
underlying the system of equations. It is exactly this hyperbolic structure
which makes it possible to circumvent the derivative loss and propagate high
enough Sobolev norms of the solution. The well-posedness result is local in
time; two continuation criteria are also established.
