  We discuss the Cauchy problem for a system of semilinear wave equations in
three space dimensions with multiple wave speeds. Though our system does not
satisfy the standard null condition, we show that it admits a unique global
solution for any small and smooth data. This generalizes a preceding result due
to Pusateri and Shatah.
  The proof is carried out by the energy method involving a collection of
generalized derivatives. The multiple wave speeds disable the use of the
Lorentz boost operators, and our proof therefore relies upon the version of
Klainerman and Sideris. Due to the presence of nonlinear terms violating the
standard null condition, some of components of the solution may have a weaker
decay as $t\to\infty$, which makes it difficult even to establish a mildly
growing (in time) bound for the high energy estimate. We overcome this
difficulty by relying upon the ghost weight energy estimate of Alinhac and the
Keel-Smith-Sogge type $L^2$ weighted space-time estimate for derivatives.
