YOU SAID:

I broke my plate after I broke my neighbor's plate because my neighbor broke both his and my plates

INTO JAPANESE

隣人がプレートを壊したので、隣人のプレートを壊した後、プレートを壊しました

BACK INTO ENGLISH

My neighbor broke the plate, so after breaking the neighbor's plate, I broke the plate

INTO JAPANESE

隣人がプレートを壊したので、隣人のプレートを壊した後、プレートを壊しました

BACK INTO ENGLISH

My neighbor broke the plate, so after breaking the neighbor's plate, I broke the plate