Reddit mentions of Venstar ACC0410 Add-A-Wire Accessory for All 24 VAC Thermostats (4 to 5 Wires), White

https://www.amazon.com/Venstar-ACC0410-Wire-Accessory-Thermostats/dp/B01IF3QXMC


something like this, that mystically adds a wire perhaps?

Buy the add-a-wire and you will then have a common wire. Easy to do, trust me.

Venstar ACC0410 Add-A-Wire Accessory for All 24 VAC Thermostats (4 to 5 Wires), White https://www.amazon.com/dp/B01IF3QXMC/ref=cm_sw_r_cp_apa_B9SRAbW6XM3TD

Venstar Add-A-Wire Adapter

https://www.amazon.com/Venstar-ACC0410-Wire-Accessory-Thermostats/dp/B01IF3QXMC/ref=sr_1_1?ie=UTF8&qid=1505259903&sr=8-1&keywords=Venstar+Add-A-Wire

It's hard to tell from the picture, but on the control panel side, you have the cable that splits out to have its red wire combined with that yellow wire on the wire nut.

That cable looks like your thermostat cable. I can't tell for certain, but is there a fifth, possible blue wire wrapped around the sheathing of the cable? If that isn't what I am seeing, is there a fifth cable in that bundle?

If so, you should be about to connect that wire to the C terminal on your control board. Back at the thermostat, dig out the hole the wires come through and see if you can't pull a little extra slack of the cable out. Depending on how your house is setup, you might even check where the cable runs in the attic and undo a staple or two if needed to create some slack. Strip the end of the fifth wire and there is your c connection.

If I'm seeing wrong and there isn't a fifth wire, your best bet is likely one of these


https://www.amazon.com/gp/aw/d/B01IF3QXMC/

Venstar ACC0410 Add-A-Wire Accessory for All 24 VAC Thermostats (4 to 5 Wires), White https://www.amazon.com/dp/B01IF3QXMC/ref=cm_sw_r_cp_api_i_fPY2DbGJFZZCP

Venstar Add a Wire works well if you have a 4 wire setup and want a C wire.

Venstar ACC0410 Add-A-Wire Accessory for All 24 VAC Thermostats (4 to 5 Wires), White https://www.amazon.com/dp/B01IF3QXMC/ref=cm_sw_r_cp_api_i_hbbqDbMETV109

Also Nest has a feature called Airwave that is supposed to turn off your compressor and keep running the fan but only after temp has been achieved.

As a last resort if the wire/cable is stuck and can't be replaced then you can go the add-a-wire route. You might need to drill a 1/2 hole below the current hole to stuff the diode and wire nut into.
https://www.amazon.com/Venstar-ACC0410-Wire-Accessory-Thermostats/dp/B01IF3QXMC
https://www.youtube.com/watch?v=R3pea5qtygo

If you can, stop and go open the panel on your hvac system (Attic/garage/Basement/crawlspace)

There you will find where those wires connect. take a picture, note it down.

There is a good chance that you could simply connect the C wire. There

Or use a c-wire bypass
https://www.amazon.com/Venstar-ACC0410-Wire-Accessory-Thermostats/dp/B01IF3QXMC

or

https://www.amazon.com/Honeywell-THP9045A1023-Wiresaver-Wiring-Thermostat/dp/B0060ASWPE/

Pulling a new cable using the old cable is problematic in most homes if you don't have access to the cable at every point. Either it will be stapled, or it will be passing around too many tight bends with high friction.
There are some alternatives:

1. The new thermostat doesn't necessarily have to go in the same place as the old thermostat. Maybe there's an alternative location in your house that's a lot easier to pull a wire through to, and still reasonably placed for central temperature sensing.
   
2. You can buy no-C-wire add-on solutions like the Venstar units (e.g., https://www.amazon.com/Venstar-ACC0410-Wire-Accessory-Thermostats/dp/B01IF3QXMC). They basically re-purpose one of the control lines as a C wire, and then use special signalling to create a virtual control line at a receiver that mounts behind the thermostat.
   
3. You can install an independent 24vac power supply (110v-to-24v ac transformer) closer to the thermostat. The sole purpose of the C wire is to provide 24vac power to the thermostat. You don't necessarily have to run that power all the way from the furnace/ac.

To me it sounds like you need this: https://www.amazon.com/Venstar-ACC0410-Wire-Accessory-Thermostats/dp/B01IF3QXMC

Also use this as a guide: https://youtu.be/R3pea5qtygo

Edit: Reason I say so is you should always use a common wire with Nest. You have to check your furnace terminal to see if it has a the right setup.

I have had a very similar experience to you and the problem was solved by installing a C-wire between the furnace and Nest. If you do not have a spare fifth wire going between your thermostat and your furnace, you can:

• Run an extra wire
  
• Install an Add-A-Wire kit (I'm not sure how well these work but it has good reviews)
  
• Temporarily change your fan wire into a C wire (but you won't be able to manually turn your fan on)