Combining indiscernible sequences

Question

Fix some structure $M$ and $C\subseteq M$. Suppose $(a_i)_{i<\omega}\subset M$ and $(b_i)_{i<\omega}\subset M$ are $C$-indiscernible sequences. Can we conclude that $(a_ib_i)_{i\leq \omega}$ is $C$-indiscernible sequence as well?

Answer 1

Here I'm going to answer my question based on Alex Kruckman's hint in the comments.

The answer is NO. Here is a counterexample!

In this picture $(a_i)_{i<\omega}$ and $(b_i)_{i<\omega}$ are indiscernible sequences but $\text{tp}(a_0b_0)=\text{tp}(a_0)\neq \text{tp}(a_1b_1)$.