Consider the ring $$ R = (\mathbb{Z}/2)[y_0,y_1,y_2,\dotsc]/ (y_0^3+y_0y_1,\;y_1^3+y_1y_2,\;y_2^2+y_2y_3,\dotsc). $$ This is also generated by the elements $x_n=\sum_{i\leq n}y_{n-i}^{2^i}$. It can be shown that the nilradical is generated by the elements $x_nx_m$ with $n\neq m$, and that it is not finitely generated. One can also give $R$ a grading with $|y_i|=2^i$. It can be shown that $R$ is coherent in the graded sense: any homogeneous ideal with a finite set of homogeneous generators has a finite homogeneous presentation. I am not sure whether it is coherent as an ungraded ring. All of this is in the paper "Large self-injective rings and the generating hypothesis", by Leigh Shepperson and myself (Theorem 1.6).

Consider the ring $$ R = (\mathbb{Z}/2)[y_0,y_1,y_2,\dotsc]/ (y_0^3+y_0y_1,\;y_1^3+y_1y_2,\;y_2^3+y_2y_3,\dotsc). $$ This is also generated by the elements $x_n=\sum_{i\leq n}y_{n-i}^{2^i}$. It can be shown that the nilradical is generated by the elements $x_nx_m$ with $n\neq m$, and that it is not finitely generated. One can also give $R$ a grading with $|y_i|=2^i$. It can be shown that $R$ is coherent in the graded sense: any homogeneous ideal with a finite set of homogeneous generators has a finite homogeneous presentation. I am not sure whether it is coherent as an ungraded ring. All of this is in the paper "Large self-injective rings and the generating hypothesis", by Leigh Shepperson and myself (Theorem 1.6).