Partition regularity dichotomy under an extra constraint

For a partition regular matrix $A$ and any property $P$ of its positive solution vectors, either every finite coloring admits a monochromatic solution with $P$, or every finite coloring admits one without $P$. If the first alternative fails, take a witnessing coloring and pair it with any other coloring. Partition regularity of the product coloring forces a solution avoiding $P$ in the other coloring. The alternatives need not be exclusive.