What is an equivalent definition of mP/poly in terms of a Turing machine? P/poly is the class of decision problems solvable by a family of polynomial-size Boolean circuits. It can alternatively be defined as a polynomial-time Turing machine that receives an advice string that is size polynomial in n and that is based solely on the size of n.

mP/poly is the class of decision problems solvable by a family of polynomial-size monotone Boolean circuits, but is there a natural alternative definition of mP/poly in terms of a polynomial-time Turing machine?
There is a notion of a monotone non-deterministic and, more generally, alternating Turing machine in the paper Monotone Complexity by Grigni and Sipser. Since polynomial time is the same as alternating logarithmic space, a machine characterization of uniform $\mathsf{mP}$ is the monotone alternating logspace Turing machine. Providing such a machine with polynomial advice will then give a machine definition of $\mathsf{mP/poly}$.