How can Forex Realized PnL be calculated when trading a pair of two foreign currencies?

You can't really. You have balances in 3 currencies. That is what your actual holdings are and what you manage.

You can find a notional P&L if you use a notional rate to convert your balances in the non USD currencies into USD. That gives you a single number which you can use as an estimate of how well you are doing, but remember you still have risk, USD might drop immediately against other currencies so you lose a lot.

You close a position by actually executing a trade to make one currency's balance become 0. This leaves an amount in the other currency and that is your P&L in that currency.

If you need a figure for accounting or tax purposes you will need to ask your accountant or tax authority how to calculate your P&L.

As a note (for simplicity let's assume that all three exchange rates (EUR/GBP, EUR/USD, GBP/USD) have moved in my favour), - that will never happen as in practice the EUR/GBP will be kept in step with the others by arbitrage by the big banks and so all cannot increase at once.

Forex can be accounted by accepting the account balance. For example the year beginning account balance is 1000, the total deposits are 200, the total withdrawals are 100, the year ending account balance is 1200, and then the account gain is 1200 - 1000 - 200 + 100 = 100 .

Or the software that I develop can update an increase in account balance with a dividend operation and update a decrease in account balance with an expense operation but with the deposits and withdrawals already input.

Now, straight from my files, here is an example of accounting a EUR/JPY currency pair in a USD account:

EUR/JPY is 122.148 (as 1 EUR buys 122.148 JPY)

USD/JPY is 107.757

[1 / 107.757] = [X / 122.148] with X = 1.1336

and then

1.1336 USD buys 122.148 JPY .

Also

EUR/USD is 1.1336 and so that number could have just been directly pulled to replace 1 EUR buying 122.148 JPY with 1.1336 USD buying 122.148 JPY.

Think not about positions, but about the value of the assets in your portfolio in your home currency.

It suffices to know the rates in USD, e.g. let Rx = EUR/USD, Ry = GBP/USD, then assuming the "no arbitrage principle", we have EUR/GBP = Rx/Ry. Indeed, since the only thing your tax authority cares about is the value of your assets in USD, whether the "no arbitrage principle" holds is irrelevant, since the EUR/GBP rate is irrelevant as far as the taxman is concerned.

Let x0 and y0 be the balances of EUR and GBP held, respectively, at time t0. Let Rx0 and Ry0 be the values of EUR/USD and GBP/USD at time t0. To clarify, in this way, if you held zero EUR and zero GBP, and then went long on EUR/GBP as you describe (i.e. going long on EUR/USD and shorting GBP/USD), we would have x0 > 0, and y0 < 0.

Similarly, let x1, y1, Rx1, Ry1 be the same quantities at a later time t1.

Then the USD equivalent of the amount of EUR held at time t0 is x0 * Rx0, and likewise, the USD equivalent of the amount of GBP held at time t0 is y0 * Ry0. Thus, the value of our portfolio in USD at time t0 is x0*Rx0 + y0*Ry0.

Similarly, the value of our portfolio in USD at time t1 is x1*Rx1 + y1*Ry1.

Thus, the profit realised from time t0 to time t1 is

(x1*Rx1 + y1*Ry1) - (x0*Rx0 + y0*Ry0).