Let V be an irregular variety defined by a set of regular identities and an identity of the form \(t(x,y)=t(x,z)\). (Any irregular variety can be defined in such a way.) A variety \(\bar V\) containing the regularization of V is introduced and studied. It has the following properties. \(\bar V\) is finitely based if V is. Every algebra from \(\bar V\) is a coherent Lallement sum (this is a generalization of a Płonka sum) of algebras from V. There is a theorem characterizing the structure of algebras in \(\bar V\) by means of some special coherent Lallement sums. Some sufficient conditions are given implying that the regularization of an irregular variety consists of Płonka sums only. The last section deals with equational bases for regularized varieties.