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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 allultraimaginaries bounded over $X$. In particular, we sharpen a result of Wagnerto 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 overhyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ andultraimaginaries $b$ 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 sequenceover $A$ if and only if whenever $I$ and $I'$ have the same Lascar strong typeover $A$, $I$ and $I'$ are related by the transitive, symmetric closure of therelation '$J+K$ is $A$-indiscernible.' This is also equivalent to $I$ being'based on' $A$ in a sense defined by Shelah in his early study of simpleunstable theories.  Finally, we show that for any $A$ and $b$ in any theory $T$, if there is anErd\"os cardinal $\kappa(\alpha)$ with $|Ab|+|T| < \kappa(\alpha)$, then thereis a total $\hspace-5mu\indbu$-Morley sequence $(b_i)_{i<\omega}$ over $A$ with$b_0 = b$.

Bounded ultraimaginary independence and its total Morley sequences