How to measure power with a CT Clamp on a split phase system

Here in the US we have a single phase split power system (120v/N/120v). I want to measure the total power consumed by my house.
Hooking up a CT Clamp to the neutral main wire will only give me the consumption of the 120v appliances, all the big appliances using 240v do not use the neutral wire, so they would not be measured.
If I hook up 2 CT Clamps to the mains, one on each phase, and add the power of the 2 together, it will also show me the correct consumption for the 120v appliances, but it would double the consumption on the big appliances, because the same current will be sensed by both clamps.
What is the correct way of measuring power consumption on a US split phase system?

You need a CT on each of your two phases.

The CT should be marked with a direction arrow or polarity. Install them accordingly.

Have not done this before, but in principle should be similar to this install - starting around 2:48
How to Monitor Your Home’s Energy Usage | Sense Energy Monitor Install & Review - YouTube

or 2:22 of this one:
Which Smart Energy Monitor Is Right For You? ShellyEM vs Sense - YouTube

The currents in Ampere that you measure this way are correct. After all you measure and display the currents in each main, not in a single consumer device.

The fact that these currents may run in and out of the same load is no contradiction to that measurement. You do NOT measure the current through one load but through each main of your supply.

In the end you usually want to derive the consumed power from the mains current. This is done by multiplying the current in each main by 120 V. (Ignoring Cos Phi for a moment - see below.) The power that you get by this multiplication is also still correct for your 240 V devices. For a single 240 V load you will get twice the current in the calculation, but then you multiply it with only 120 Volts instead of 240. So the end result remains correct.

About Cos Phi:
To calculate the power P from the current I and the voltage U you also need the cos phi (“power factor”).
This is the phase shift between the sine wave of the current and the sine wave of the voltage.

P = U * I * cos phi

For normal resistor loads cos phi is 1 and you can ingore it. This is true for many heating loads like ovens and radiators. But it is not true for inductive and capacitive loads like motors, transformers and most electronic devices.
With clamps alone you cannot measure the power factor. Instead you need more complex sensors that measure current AND voltage and the phase shift between them. Have a look at ATM90E32 Power Sensor for an example.

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Thanks everyone for your replies.
I’m aware that I need to measure voltage and phase shift, I just did not see any way of doing this directly on ESPHome. Maybe my question was not phrased correctly.
I think I’m just going to do this with an Arduino using emonLib, and pass the info back to the ESP via serial.

I think you are asking if an esp 8266/32 can measure AC voltage directly, that answer is no. You’ll need another component. Any component marked voltage is a means of measurement.

Did you read the very last sentence of my posting?

I normally do 1 loop of one of the split phases so that it goes backward in the clamp. This way the currents adds up and you can measure both with 1 CT clamp.
Alternatively the 2 CT clamps on the mains also works, big appliances use both lines and twice the power for the same current.