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Julie here from Shopify Support. Thanks for reaching out.
It sounds like you've set up email forwarding, is that right? With email forwarding, customers can email you at your custom email address (firstname.lastname@example.org) and those emails are automatically forwarded to an existing email address (in this case, your Gmail address).
However, if you want to reply to a customer email, then you must do so directly from your Gmail address, rather than the custom email address. Because you'll be responding from your existing Gmail account, this is the email address that will appear in your customer's inbox.
If you don't want customers to see your Gmail address in your replies, then you'll just need to use a third-party email hosting service. With third-party email hosting, you'll be able to send and receive emails from a custom email address associated with your domain. Not only will this make your replies look more professional, but having a separate inbox for your business can make things much easier to manage.
Both Zoho Mail and Google Workspace are popular third-party email hosting services and they integrate directly with Shopify. If you decide to use another email hosting service, then you can still connect it with Shopify; in this case, you'll need to add an MX record to connect it to your domain by editing your DNS settings.
I hope this helps! Have you considered using third-party email hosting? Aside from communicating with suppliers and stockists, are you planning on doing any email marketing?