Example of product cipher more secure than its components
For


a $1$-bit message $m$
a randomly generated, fixed prime $k$ (the key)
per-ciphertext randomly generated $r$ and/or $e$


C1, C2

$$C_1 : (k * r) + m$$
$$C_2 : (2 * e) + m$$

A1, A2

$$\begin{align}A_1(c_0, c_1) : c' = c_0 - m_0\\c'' = c_1 - m_1\\k = \operatorname{gcd}(c', c'') = \operatorname{gcd}(k*r_0, k*r_1)\end{align}$$
$$\begin{align}A_2(c_0) : m = c_0 \bmod 2\end{align}$$

C12

$$C_{(12)} : (k * r) + (2 * e) + m$$
$$A_{(12)} : \text{Solve the AGCD problem}$$

Notes


Attacks assume the known-plaintext attack scenario
$C_2$ by itself might be considered a pathological example by itself, but as you can see above, when combined with some $C_1$ it contributes immensely to security.
$A_1$ might require multiple applications if $k$ is not prime