I was told there is a Court of Criminal Appeals case decision saying that allowing the jury alternates to deliberate with the jurors constitutes reversible error. Anyone have this citation?

I don't know a case cite, but I know it is improper for any person not of the jury to be present during deliberations. And an alternate is not a juror until a sitting juror has been dismissed and the alternate becomes a juror.

Every court I've been in discharges the alternates prior to the beginning of deliberation, and of course they are not allowed to further participate in the case.

Are you thinking of Trinidad v. State, 2008 WL 2916307? It's a San Antonio case, but it means what I think you think it means.