So this lagrangian was used as an example for deriving the equations of motion using the Euler-Lagrange equations in our lecture notes.
$$ L (\phi )=-\phi (x,t)^{2}+m\left(\frac{\partial \phi }{\partial x}\right)^{2}-c\frac{\partial \phi }{\partial t}\frac{\partial \phi }{\partial x}$$
but im wondering if it describes any "real" system or if its just some generic lagrangian conjured up for this specific example
This is just the usual Klein-Gordon Lagrangian in a weird set of coordinates. Specifically, if we define $$ \tau = \frac{2 \sqrt{m}}{c}t + \frac{1}{\sqrt{m}} x , \qquad \xi = \frac{1}{\sqrt{m}} x, $$ then we have \begin{align*} \frac{\partial \phi}{\partial x} &= \frac{\partial \xi}{\partial x} \frac{\partial \phi}{\partial \xi} + \frac{\partial \tau}{\partial x} \frac{\partial \phi}{\partial \tau} = \frac{1}{\sqrt{m}} \left[\frac{\partial \phi}{\partial \xi} + \frac{\partial \phi}{\partial \tau} \right] \\ \frac{\partial \phi}{\partial t} &= \frac{\partial \xi}{\partial t} \frac{\partial \phi}{\partial \xi} + \frac{\partial \tau}{\partial t} \frac{\partial \phi}{\partial \tau} = \frac{2 \sqrt{m}}{c} \frac{\partial \phi}{\partial \tau} \end{align*} and so \begin{align*} m\left(\frac{\partial \phi }{\partial x}\right)^{2}-c\frac{\partial \phi }{\partial t}\frac{\partial \phi }{\partial x} &= \left[\frac{\partial \phi}{\partial \xi} + \frac{\partial \phi}{\partial \tau} \right]^2 - 2 \frac{\partial \phi}{\partial \tau} \left[\frac{\partial \phi}{\partial \xi} + \frac{\partial \phi}{\partial \tau} \right] = \left(\frac{\partial \phi}{\partial \xi} \right)^2- \left(\frac{\partial \phi}{\partial \tau} \right)^2. \end{align*} I kind of doubt that this was your instructor's intention, but there's not a lot of freedom in writing out second-order kinetic terms, and it turns out there are only a few possible "canonical forms" such as this one.
