Let $A_1,\dots,A_n$ be matrices, with no row or column of $0s$, and such that for every $i=1,\dots,n$ there does not exist a decomposition of $A_i$ of the form $$ A_i = \oplus_{j=1}^n B_j \qquad (\exists j_1\neq j_2)\;(\exists k \in \mathbb{R}) \,B_{j_1}=kB_{j_2}, $$ where the direct sum of matrices is defined here.

Is there a reasonable compatibility criterion on the $A_i$ such that $A= \oplus_{i=1}^nA_i$ also does not admit such a decomposition; i.e.: there do not exist $C_1,\dots,C_t$ such that $$ A = \oplus_{i=1}^t C_i \mbox{ and } (\exists t_1\neq t_2)\;(\exists k \in \mathbb{R}) \,C_{i_1}=kC_{i_2}. $$

**Non-example:**
An interesting non-example, which illustrates part of the issue is this:
$$
A_1\triangleq \begin{pmatrix}
1 & 1 & 0\\
1 & 1 & 0\\
0 & 0 & 1
\end{pmatrix}
A_2 \triangleq (2),
$$
then $A_1,A_2$ are distinct and of distinct size but, $B_1$ is the 2x2 matrix of 1s, $B_2=(1)$, $B_3=A_2$ gives the decomposition I don't want.