You drafted with Claude. They read with ChatGPT, or with nothing. Somebody on your team will treat that as a blocker — "we should all be on the same tool" — as if correspondence required a matching runtime.
It does not. Two people have never needed the same word processor to exchange a letter. They need a document both can open, and a rule about who may put one in the other's mailbox.
The package is the common object
Assistants can talk to each other because neither of them is the channel. The channel is a briefing: an ask, a brief, labels, a note you approved. Their model pulls it the same way yours would. If they have no model, they open the page. The bytes do not care which vendor sat on either end.
What fails is trying to share the session. A ChatGPT link is a window into your chat. Their Claude cannot treat that as inbound mail. There is no ask, no consent gate, no label that says which sentences you signed. You have shown them your scratchpad and hoped their assistant would invent a job.
Consent is still the gate
Once they have an activated account, you do not get to drop packages into their assistant because you found their address. AI-to-AI communication needs consent. An accepted contact pair, or the send is refused. The refusal does not say whether they are on RelayLink, so the gate is not a directory lookup.
Different assistants do not weaken that. If anything they make it more important. You are not only writing a person. You may be writing a model that will summarise you, act on you, and speak to them in a voice that is not yours. They chose that model. They did not choose you until they accepted.
A first send to someone with no account is still email. They can stay off the product forever and still reply. Matching stacks was never the on-ramp.
Two assistants on one person is a different case
If you use Claude at work and ChatGPT at home, that is not a cross-company problem. That is one account and a hand-off to yourself. Address the package to your own email. The other model sees the words you approved, not an export of the first chat. No second account. No contact request with yourself.
Do not confuse that loop with writing another person. Their inbox is not yours. Their model is not a second window on your account.
What you stop negotiating
You stop waiting for them to "just try Claude." You stop pasting vendor-specific share links into email and calling it a brief. You stop treating a mismatched stack as a reason to book a call.
You write the package, you approve it, you send it. They read it with whatever they have. If they want their assistant to draft the reply, they can. If they want to type, they can. The correspondence does not belong to either vendor. It belongs to the two of you.