  We show that the properties of Lagrangian mean curvature flow are a special
case of a more general phenomenon, concerning couplings between geometric flows
of the ambient space and of totally real submanifolds. Both flows are driven by
ambient Ricci curvature or, in the non-K\"ahler case, by its analogues. To this
end we explore the geometry of totally real submanifolds, defining (i) a new
geometric flow in terms of the ambient canonical bundle, (ii) a modified volume
functional which takes into account the totally real condition. We discuss
short-time existence for our flow and show it couples well with the
Streets-Tian symplectic curvature flow for almost K\"ahler manifolds. We also
discuss possible applications to Lagrangian submanifolds and calibrated
geometry.
