Suppose $A$ be a skew-symmetric matrix whose entries are positive numbers. A perturbation of $A$, $A'$, is obtained by adding another skew-symmetric matrix whose entries are positive integers.

My question is for all such $A$ is there any perturbation $A'$ such that all Pfaffian minors of $A'$ are positive?

By definition a Pfaffian minor of a skew symmetric matrix $A$ is a Pfaffian of a submatrix of $A$ consisting of rows and columns indexed by $i_1, i_2, ..., i_{2t}$ for some $i_1 < i_2 < ... < i_{2t}$.