Asking that the metric of a complex Finsler space should be strong convex, Abate and Patrizio [1] associate to the real tangent bundle a real Finsler metric for which they analyze the relation between Cartan (real) connection of the obtained space and the real image of Chern-Finsler complex connection. Following the same ideas, in the present paper we shall deal with the more general case of a complex Lagrange space $(M,L)$. As distinct from these authors, we shall associate to the Hermitian metric $g_{i\bar j}(z,\eta)$ of a complex Lagrangian $L$ its real representation $\stackrel{R}{g}_{ab}(x,y)$. The obtained real space $(M,\stackrel{R}{g}_{ab})$ is a generalized Lagrange space [10]. Furthermore, the possibility of its reduction to one real Lagrange space, in particular the Finsler one, is studied. A comparative analysis of the elements of Lagrange geometry [10]: nonlinear connection, $N$-linear connection, metric canonical connection, and so on, and their corresponding real image from the complex Lagrange geometry [11] is made.