jmc

We can solve for $y,$ to get $$y=\frac{4-x-z}{2}.$$Substituting, we get $$xy+xz+yz=\frac{-x^2+4x-z^2+4z}{2}=\frac{8-(x-2{)}^2-(z-2{)}^2}{2}.$$The maximum value is then $\boxed{4},$ which occurs when $x=2$ and $z=2$ (and $y=0$).