I think a few people have hit on the problem, but no one has really given you the most effective way of fixing it. Your system is essentially ok...it appears that the only thing you are missing is a high pass on the cabin speakers.
By sending bass frequencies to your other speakers, you are causing them to distort.
There are two cheap and effective ways of keeping the bass out of your cabin speakers...then there are the expensive ways. If cash is tight, go with the cheap way, if you have some money to spend there are a few things you can upgrade:
Cheap ways:
1. Buy or build some passive crossovers. These are sold under the name "bass blockers" Essentially they are a capacitor placed across the terminals on the driver (someone correct me if I'm wrong....). You put this between the output on your amp and the speakers. This is the cheapest way to go about it. Unfortunately the slope is really shallow on the crossover (-6dB/oct) but it will keep you from distortion...I would shoot for around 80 for the rears, and 100 Hz for the fronts. Total cost $20-40
2. Buy an electronic crossover. MTX made one a while back (I think they still do) that is around $50 and is a two channel and they made a three channel for around $80. One of the two....again I cna't remember...it's been a tough day....accepts line level inputs. You can use that to highpass the soundstream, set the gains, and control the other MTX amp. I have one that is in my other car....It's a fairly good unit for the price, and will do what you need for a tiny bit of cash. Total cost $20-80 (look on ebay - I sold a really sweet PG MX3i for under 50 in mint condition not too long ago)
Expensive ways:
1. New Head: Like a few people touched on, some of todays headunits have crossover networks built in, along with sublevel control...this would be handy in your situation...it will provide you with RCA outputs for your amps, and will have a lot more features then your stock head. Cost: $150-1000 depending on what you want
2. New Amp: you could ditch that soundstream, and get a 4 channel amp to power the cabin speakers. Most all new amps have built in crossovers. Total cost: $100-2000