Rear windshield washer motor works, but no fluid comes out

"If the windshield wiper fluid of your Chevy Equinox ends up on the rear window instead of the windshield, there could be several possible causes:

Damaged or incorrectly connected windshield wiper fluid tube: The tube that carries the windshield wiper fluid from the reservoir to the base of the front windshield wiper may be damaged or not connected properly, causing the fluid to leak elsewhere.

Clogged windshield wiper fluid outlet: The outlet of the windshield wiper fluid on the base of the front windshield wiper may be clogged with dirt or debris, causing the fluid to flow onto another part of the car, such as the rear window.

Windshield wiper system issues: There may be a problem with the windshield wiper system of your Chevy Equinox that causes the fluid to leak onto other parts of the car instead of the windshield."

I'd have to pull a wiring diagram to say for sure, but it appears both the front and back wiper motors are part of the same "series" circuit, so if you disconnect one, the other will not work.

Plug the front motor back up really quick. If the rear wiper motor works, you've found the problem. Is there a particular reason why you disconnected the rear motor?

The washer fuse should have power (battery voltage) while the ignition switch is in the RUN position. Verify both sides of the fuse have power. If the rear wash functions them it's not the fuse.

The switch signals the Body Control Module (BCM). The BCM energizes the front washer relay.

Input to BCM from switch is BCM connector C1,24 PK. Output from BCM to relay is connector C4 15 OG.

You need to verify continuity from the the switch to the BCM and from the BCM to the relay.

Comment below for further assistance.

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How To Install Replace Rear Window Wiper Motor 2000 06 Chevy Suburban Tahoe

There is a electric motor that pumps washer fluid . Located in the bottom of the washer fluid tank .
Wash Operation (Front and Rear Wipers)
The front and rear washer systems use the same windshield washer fluid pump. When either the front or rear wiper is in the washer position voltage will be supplied to one side of the windshield washer fluid pump (depending on whether it is front or rear), while the other side of the pump is grounded.
At this point, the windshield washer fluid pump sprays solvent onto the windshield. When the front or rear washer is activated, the BCM receives an input from the switch that is activated. The BCM will control the wipers while in the WASH mode. The BCM will control either the front wiper relay or the rear wiper relay to turn on the wipers. The windshield wipers operate at low speed for 3 swipes and returns to the park position.

When turning on the washers do you hear the motor run ? Have some one try it while you listen at the washer fluid tank ! These pump motors do go bad .
turn OFF the ignition.
Disconnect the harness electrical connector of the windshield washer fluid pump.
Connect a test lamp from the windshield washer pump control circuit to a good ground .
Turn ON the ignition, with the engine OFF.
Press the front washer switch.
Does the test lamp illuminate?
YES or NO !

It sounds as if the windshield washer pump, located at the base of the windshield washer fluid reservoir, has lost it's seal. When this occurs, washer fluid seeps into the pump and shorts it out. I advise that you remove the pump to confirm this before replacing the fuse. A replacement wind washer pump and fuse can be found at your dealer, local auto parts store or the internet (Rockauto.com has always been a great source for quality parts at a very competitive price and I use them all the time). Hope this helps.

The hose could be pinched some where along the line or there could be something plugging it.  I once found a holder in the back of one of my own vehicles that held washer fluid.  I found it by accident. There could be one there, and it could be covered, worth the look, I think.

the pump has packed up its under the inner wing on the washer bottle and its one pump that does both by reversing the motor ,its a job to get too without removing the plastic inner wing

Your rear bottle for the wipers should be at the back of the vehicle.Look in back for removable panel and bottle should be there.Turn your ignition on(no start)and activate rear washer Listen for the sound of the motor