Staz when a coy is born it has its own register of shareholders,...

Staz when a coy is born it has its own register of shareholders, the so-called issuer register which issues shareholding statements to its new shareholders.

When a coy dies it comes off the CHESS register which means it is no longer an ASX qualifying stock, in simple terms. The shares still exist so the coy transfers the data to its own register again, the issuer register.

Alpha and Omega.

If you want to avoid having to wait years to claim a tax deduction - waiting for auditors and closure professionals to value remaining assets and pay off creditors before finally burying the coy, I'd recommend selling to somewhere like Delisted .com. I say sell but actually you'll pay them to take the shares - about $100-$150, and then you can claim the tax deduction as you will have legally disposed of them. Worth it in my books.
Not financial advice.