From what I've read, the thermostat will get voltage via the red wire when the system powers up, turn on the system when it needs power, or take some power from the batteries if it needs it and causes the batteries to drain quicker. The C wire eliminates this.
From the 5 wire running between the Tstat and furnace.
Red is 24v
White is 1st stage heat
Green is Fan
Yellow is 1st stage cooling.
Blue would be for 2nd stage cooling... using it as a common is fine.
Your condensing unit is probably wired incorrectly.
From the terminal strip where the yellow is tied down, the red from the 2 wire should be tied down.
The white should be tied down on the far right where the red is now.
Just opposite of what you have.
I know, the unit is working fine.
Why does it make a difference?
If the condensing unit is fused, you are putting the fuse after everything it is supposed to protect.
Anything that shorts out can blow your transformer.
Also, anyone working on the unit will except the red to be hot.
That is the wire you would disconnect first when working on the unit.
Could cause the tech to short something out.
Before you change anything... make sure you check the wiring in the condensing unit to verify.