The reason for this is something called "tire load sensitivity." When you stiffen the bar and increase the vertical force a bunch on one tire, the lateral force goes up, yes, but the friction coefficient of that tire goes down.
In physics 101 it's taught that Coulomb friction is how everything works. I.e, the friction coefficient (let's call it "mu") is constant between two surfaces, such as a chunk of wood on smooth tile. This is pretty close to true when dealing with hard materials, but it's not correct for tires or other things made of rubber even on hard surfaces.
Imagine the two rear wheels. If Coulomb friction actually held true for tires and their friction coefficient (mu) with the road was 1, the amount of side force each of the tires could produce would be the same as the vertical force or load on them. So in a case where you have little weight transfer (your soft rear ARBs) on an axle with a weight of 1000, you might have vertical loads like this:
Inside tire 400
Outside side 600
The maximum cornering forces are also 400 and 600 (multiply the vertical load by mu, in this case 1). So total cornering force at the rear is 1000. If you then stiffen the rear end so you get a lot more weight transfer, you might have vertical loads like this:
Inside tire 100
Outside tire 900
If mu stays at 1 then you still have the same overall cornering force on that axle of 100 plus 900, or 1000. So really, if tires worked that way ARB's wouldn't do much of anything at all to the handling balance.
Tires have that "load sensitivity" mentioned earlier. This is just a fancy phrase meaning that the friction coefficient (mu) varies with the vertical force. I.e., it doesn't stay constant at 1 or some other number all the time. If you increase the vertical force, mu drops. The cornering force for that tire will of course increase, yes, but it does not increase in proportion to the vertical force increase. If you double the vertical load, the cornering force almost doubles too, but not quite. The inside tire produces less force since it's less loaded, but produces just a bit more than it otherwise would since its friction coefficient goes up. However, the outside tire loses more than the inside tire gains, so in the end you have less overall cornering force on that axle. Harder springs or ARBs will then make things more slippery on that axle.
If mu changed with load something like this:
Load
100 1.12
400 1.08
600 1.05
900 1.00
And we look at our soft and hard rear ends again (that doesn't sound right
) we will have different overall cornering forces for each one.
Soft:
Inside tire 400 load means mu = 1.08, so cornering force for this tire is 400*1.08=432
Outside side 600 load means mu = 1.05, so cornering force for this tire is 600*1.05=630
Total cornering force at the rear with the soft rear setup is 630 + 432 = 1062.
If we switch to the hard anti-rollbar we have a different situation because of this load sensitivity:
Inside tire 100 load means mu= 1.12, so cornering force for this tire is 100 * 1.12 = 112
Outside tire 900 load means mu = 1.00, so cornering force for this tire is 900 * 1.00 = 900
That gives us a cornering force of 1012 at the rear (900+112). This is actually less than the 1062 we get with the softer set up, so we tend to get more oversteer when we stiffen up the rear.
This phenomenon is pretty much the only thing that makes chassis tuning at the limit possible at all in most cars