What Makes the Study of Non Strictly Hyperbolic PDEs Harder?

Question

A non-constant coefficient partial differential equation of order $m$ of the form $P(x,\tau, D_x / i)$ is called strictly hyperbolic if all roots of the homogeneous polynomial $P_m(x,\tau, \xi)$ in the $\tau$ variable are distinct and imaginary for all $x$. On the other hand, the equation is just called hyperbolic if the roots are just imaginary (but not necessarily distinct). What properties do strictly hyperbolic partial differential equations have that make it necessary to restrict to the study of strictly hyperbolic equations. I am reading some notes on Fourier integral operators and the motivation for this restriction isn't obvious since I don't have a very broad knowledge of partial differential equations.