With this implementation, the checkout process is unchanged and it's not possible for us to change the way Checkout page process tax as we do not have access to it in a basic Shopify store account.

We can only show an extra tax included price below the normal price (indicating it's tax included price is for only EU country).

The checkout part is out of our control as we do not have access to the Checkout page due to the limitation of Shopify itself. Only Shopify Plus accounts have access to Checkout changes, and even those access allows very limited control to what can be done.

The taxes part at checkout can only be controlled through settings > Taxes which is linked here, which can be set up at your end. Within our service scope, we can only offer to show the price with or without tax, but not the outcome at the Checkout page.

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