  We investigate a new class of $\eta$-deformed $AdS_5 \times T^{1,1}$
backgrounds produced by $r$-matrices that satisfy the modified classical
Yang-Baxter equation [Jour. High Ener. Phys. 03 (2022) 094]. We examine the
classical phase space of these (semi)classical strings by numerically studying
the dynamics of the string sigma models over this deformed background, and we
compute several chaos signals. These involve figuring out the Poincar'e section
and computing the Lyapunov exponents. In the (semi)classical limit, we discover
evidence that supports a non-integrable phase space dynamics.
