  In this paper, we develop a method to obtain the algebraic classification of
compatible pre-Lie algebras from the classification of pre-Lie algebras of the
same dimension. We use this method to obtain the algebraic classification of
complex $2$-dimensional compatible pre-Lie algebras. As a byproduct, we obtain
the classification of complex $2$-dimensional compatible commutative
associative, compatible associative, and compatible Novikov algebras. In
addition, we consider the geometric classification of varieties of cited
algebras, that is the description of its irreducible components.
