  We investigate the following model-theoretic independence relation:
$\def\indbu{{\rlap{\hspace11.9mu\vert}\lower7.5mu\smile}^{\!\mathrm{bu}}} b
\indbu_A\hspace3mu c$ if and only if $\mathrm{bdd}^u(Ab)\cap \mathrm{bdd}^u(Ac)
= \mathrm{bdd}^u(A)$, where $\mathrm{bdd}^u(X)$ is the class of all
ultraimaginaries bounded over $X$. In particular, we sharpen a result of Wagner
to show that $b \indbu_A\hspace3mu c$ if and only if $\langle
\mathrm{Autf}(\mathbb{M}/Ab)\cup\mathrm{Autf}(\mathbb{M}/Ac) \rangle =
\mathrm{Autf}(\mathbb{M}/A)$, and we establish full existence over
hyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ and
ultraimaginaries $b$ and $c$, there is a $b' \equiv_A b$ such that $b'
\indbu_A\hspace3mu c$). Extension then follows as an immediate corollary.
  We also study total $\hspace-5mu\indbu$-Morley sequences (i.e.,
$A$-indiscernible sequences $I$ satisfying $J \indbu_A\hspace3mu K$ for any $J$
and $K$ with $J + K \equiv^{\mathrm{EM}}_A I$), and we prove that an
$A$-indiscernible sequence $I$ is a total $\hspace-5mu\indbu$-Morley sequence
over $A$ if and only if whenever $I$ and $I'$ have the same Lascar strong type
over $A$, $I$ and $I'$ are related by the transitive, symmetric closure of the
relation '$J+K$ is $A$-indiscernible.' This is also equivalent to $I$ being
'based on' $A$ in a sense defined by Shelah in his early study of simple
unstable theories.
  Finally, we show that for any $A$ and $b$ in any theory $T$, if there is an
Erd\"os cardinal $\kappa(\alpha)$ with $|Ab|+|T| < \kappa(\alpha)$, then there
is a total $\hspace-5mu\indbu$-Morley sequence $(b_i)_{i<\omega}$ over $A$ with
$b_0 = b$.
