- On the iPhone, in Settings | Messages ensure that iMessage is 'On' and that you are able to Receive at your Apple ID email, as well as your phone number. Optionally turn on Send Read Receipts - it certainly enhances the experience to know that the other has read, or is typing...
- On the iPad, in Settings | Messages do exactly the same, ensuring that you're receiving messages at your Apple ID
- On each device, select Settings | General | Reset | Reset Network Settings.
For me, this combination meant that I can now type a message on either my iPhone, or iPad, and my message appears immediately on my other device. And the messages Re kept together in the same 'thread' of conversation with the recipient whereas previously I was seeing some messages associated with a phone number and others with an email address. Now the messages are unified.