Antisymmetric Table Puzzle where the Rows/Columns Sum to Zero

This can be formulated as an Integer Programming or Constraint Programming problem. Let $x_{ij} \in \mathbb{Z}$ be the value of cell $(i,j)$, then

$$\begin{align} &x_{i,j} = 0 && \text{white and blue cells}\\ &x_{i,j} \ge 1 && \text{green cells}\\ &x_{i,j} \le -1 && \text{red cells}\\ &x_{i,j}= -x_{j,i} && \forall i<j\\ &\sum_i x_{i,j} = 0 &&\forall j\\ &\sum_j x_{i,j} = 0 &&\forall i\\ \end{align}$$

These are all linear constraints. (The model can be reformulated by skipping all white/blue cells: that would make the problem smaller).

I tried your problem, and if I made no errors transcribing the data, the problem does not have a feasible solution. (Well, I made some errors. See below for updated results).

Update

After fixing my input data, and adding the objective: $$\min \sum_{(i,j) \in Green} x_{i,j}$$ I get:

The total value of the green cells is 87 (this is the optimal value: no better solution exists). This solution was found in 0.28 seconds.