Usage of Each Other

I have actually written something as below:-

They feel so warm when each other have ordered the favorite paintings for the other party.  

That's why I want to know exactly  which clause is more appropriate  to write each other has or each other have ?

I do not need another alternative way of writing the sentence.  I would just like to clarify the term EACH OTHER should consider as a pronoun of singular or plural ?

Many thanks!