  This paper studies the concepts of a totally compatible dialgebra and a
totally compatible Lie dialgebra, defined to be a vector space with two binary
operations that satisfy individual and mixed associativity conditions and Lie
algebra conditions respectively. We show that totally compatible dialgebras are
closely related to bimodule algebras and semi-homomorphisms. More
significantly, Rota-Baxter operators on totally compatible dialgebras provide a
uniform framework to generalize known results that Rota-Baxter related
operators give tridendriform algebras. Free totally compatible dialgebras are
constructed. We also show that a Rota-Baxter operator on a totally compatible
Lie dialgebra gives rise to a PostLie algebra, generalizing the fact that a
Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra.
