  Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each
real embedding $\iota$ of $F$, Hiroyuki Yoshida defined an invariant
$X(c,\iota)$ as a finite sum of log of Barnes' multiple gamma functions with
some correction terms. Then the derivative value of the partial zeta function
$\zeta(s,c)$ has a canonical decomposition
$\zeta'(0,c)=\sum_{\iota}X(c,\iota)$, where $\iota$ runs over all real
embeddings of $F$. Yoshida studied the relation between $\exp(X(c,\iota))$'s,
Stark units, and Shimura's period symbol. Yoshida and the author also defined
and studied the $p$-adic analogue $X_p(c,\iota)$: In particular, we discussed
the relation between the ratios $[\exp(X(c,\iota)):\exp_p(X_p(c,\iota))]$ and
Gross-Stark units. In a previous paper, the author proved the algebraicity of
some products of $\exp(X(c,\iota))$'s. In this paper, we prove its $p$-adic
analogue. Then, by using these algebraicity properties, we discuss the relation
between the ratios $[\exp(X(c,\iota)):\exp_p(X_p(c,\iota))]$ and Stark units.
