The constraint satisfaction problem (CSP) of a first-order theory $T$ is the computational problem of deciding whether a given conjunction of atomic formulas is satisfiable in some model of $T$. We study the computational complexity of $mathrm{CSP}(T_1 cup T_2)$ where $T_1$ and $T_2$ are theories with disjoint finite relational signatures. We prove that if $T_1$ and $T_2$ are the theories of temporal structures, i.e., structures where all relations have a first-order definition in $(mathbb{Q};<)$, then $mathrm{CSP}(T_1 cup T_2)$ is in P or NP-complete. To this end we prove a purely algebraic statement about the structure of the lattice of locally closed clones over the domain ${mathbb Q}$ that contain $mathrm{Aut}(mathbb{Q};<)$.
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