@remi This is happening more and more, providers that don't allow you to send emails from domains they are not authorutive for. We, in our email service, do the same, it is a very easy way to block potential spammers, especially for ISP's like Orange. Note that you will also have issues with SPF and DMARC, assuming those domains have their security in order.
The only proper solution is that the domain owners provide an email service for their domains you can authenticate to.
@wanwizard Sadly I don't think this is possible for large projects (ex: php.net)
And I also think this will not block a lot of spam (perhaps a few trivial ones)
@wanwizard @remi You don't even need a full service, only SMTP