10 (14 points) Positive real numbers $a, b, c, d$ satisfy $a+b+c+d=abcd$. Find
$$
a^{4}(bcd-1)+b^{4}(cda-1)+c^{4}(dab-1)+d^{4}(abc-1)
$$

the minimum value.
10. Since $a, b, c, d > 0$, and
$$
a+b+c+d=a b c d,
$$

we have
$$
a(b c d-1)=b+c+d .
$$

Similarly, we have
$$
\begin{array}{l}
b(c d a-1)=c+d+a, \\
c(d a b-1)=d+a+b
\end{array}
$$

and
$$
d(a b c-1)=a+b+c .
$$

By the AM-GM inequality, we get
$$
a b c d=a+b+c+d \geqslant 4 \sqrt[4]{a b c d},
$$

where equality holds if and only if $a=b=c=d=\sqrt[3]{4}$. Therefore,
$$
\begin{aligned}
& a^{4}(b c d-1)+b^{4}(c d a-1)+c^{4}(d a b-1)+d^{4}(a b c-1) \\
= & a^{3}(b+c+d)+b^{3}(c+d+a)+c^{3}(d+a+b)+d^{3}(a+b+c) \\
\geqslant & 12 \sqrt[12]{\left(a^{3} b a^{3} c a^{3} d\right) \cdot\left(b^{3} c b^{3} d b^{3} a\right) \cdot\left(c^{3} d c^{3} a c^{3} b\right) \cdot\left(d^{3} a d^{3} b d^{3} c\right)} \\
= & 12 \sqrt[12]{a^{9} b^{9} c^{9} d^{9}(b c d)(c d a)(d a b)(a b c)} \\
= & 12 a b c d=12(\sqrt[3]{4})^{4} \\
= & 48 \sqrt[3]{4} .
\end{aligned}
$$