  Motivated by a number of theoretical considerations, predicting the deeply
bound double-heavy tetraquarks $T^{\{bb\}}_{[\bar u \bar d]}$,
$T^{\{bb\}}_{[\bar u \bar s]}$ and $T^{\{bb\}}_{[\bar d \bar s]}$, we explore
the potential of their discovery at Tera-$Z$ factories. Using the process $Z
\to b \bar b b \bar b$, we calculate, employing the Monte Carlo generators
MadGraph5$\_$aMC@NLO and Pythia6, the phase space configuration in which the~$b
b$ pair is likely to fragment as a diquark. In a jet-cone, defined by an
invariant mass interval $m_{bb} < M_{T^{\{bb\}}_{[\bar q \bar q']}} + \Delta
M$, the sought-after tetraquarks $T^{\{bb\}}_{[\bar q \bar q^\prime]}$ as well
as the double-bottom baryons,~$\Xi_{bb}^{0,-}$, and $\Omega_{bb}^-$, can be
produced. Using the heavy quark--diquark symmetry, we estimate $\mathcal{B} (Z
\to T^{\{bb\}}_{[\bar u \bar d]} + \; \bar b \bar b) = (1.4^{+1.1}_{-0.5})
\times 10^{-6}$, and about a half of this for the
$T^{\{bb\}}_{[\bar{u}\bar{s}]}$ and $T^{\{bb\}}_{[\bar d \bar s]}$. We also
present an estimate of their lifetimes using the heavy quark expansion,
yielding $\tau(T^{\{bb\}}_{[\bar q \bar q^\prime]}) \simeq 800$~fs. Measuring
the tetraquark masses would require decays, such as $T^{\{bb\} -}_{[\bar u \bar
d]} \to B^- D^- \pi^+$, $T^{\{bb\} -}_{[\bar u \bar d]} \to J/\psi \overline
K^0 B^-$, $T^{\{bb\} -}_{[\bar u \bar d]} \to J/\psi K^- \overline B^0$,
$T^{\{bb\} -}_{[\bar u \bar s]} \to \Xi_{bc}^0 \Sigma^-$, and $T^{\{bb\}
0}_{[\bar d \bar s]} \to \Xi_{bc}^0 \bar\Sigma^0$, with subsequent decay chains
in exclusive non-leptonic final states. We estimate a couple of the decay
widths and find that the product branching ratios do not exceed~$10^{-5}$.
Hence, a good fraction of these modes will be required for a discovery of
$T^{\{bb\}}_{[\bar q \bar q']}$ at a Tera-$Z$ factory.
