Splitting a Complex FIR Filter into Real Filters
I'd say there are two ways to look at this problem.


If you're receiving a conventional $M$-QAM signal with AWGN noise, then the I and Q streams are actually two independent, real, $\sqrt{M}$-PAM signals. You can filter and process them independently. The same is true over the wireless channel, as long as the fading is flat and you do the appropriate detection in the receiver.
If $s(t)=s_I(t)+js_Q(t)$ is a complex signal and $h(t)=h_I(t)+jh_Q(t)$ is a filter's complex impulse response, then


$$s(t)\star h(t)=s_I(t)\star h_I(t)-s_Q(t)\star h_Q(t)+j(s_I(t)\star  h_Q(t)+s_Q(t)\star h_I(t)),$$

where all the filters involved are real. This follows from the convolution's properties.