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Here's a fairly obvious comment about an easy case of this problem. I post it as answer because it's too long for the comment box.

In projective terms, you are looking for the largest dimension of a linear space $\Lambda$ in $\mathbb P^{mn-1}$ whose intersection with the Segre variety $\Sigma_{m-1,n-1}$ does not have any real points.

As the answer by Sasha to the linked question on M.SE explains, for any $\Lambda$ of codimension $\operatorname{dim}\left(\Sigma_{m-1,n-1}\right)=m+n-2$, the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ has nonempty set of complex points.

Now suppose there is a $\Lambda$ of codimension $\leq m+n-2$ such that $\Lambda \cap \Sigma_{m-1,n-1}$ has no real points. Then clearly there is such a $\Lambda$ of codimension equal to $m+n-2$. I claim* that by perturbing $\Lambda$, we can arrange that the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ still has no real points, and each complex point of intersection has multiplicity 1.

Under these hypotheses, the number of complex intersection points equals the degree of $\Sigma_{m-1,n-1}$, which equals ${m+n-2 \choose m-1}$. Now complex conjugation acts on this set of intersection points with no fixed point (because a fixed point would be real). So the degree of the Segre variety must be even.

Contrapositively, if the number ${m+n-2 \choose m-1}$ is odd, then the maximal dimension of a disjoint real subspace is the same as the maximal dimension of a disjoint complex subspace, i.e. $(m-1)(n-1)$.

Unfortunately, the set of pairs $(m,n)$ satisfying this condition seems to have asymptotic density zero.

*: The set of linear subspaces intersecting $\Sigma$ transversely is a dense open set of the real Grassmannian, while the set of linear subspaces whose intersection has no real points is a Euclidean open set. So their intersection should be nonempty.

Here's a fairly obvious comment about an easy case of this problem. I post it as answer because it's too long for the comment box.

In projective terms, you are looking for the largest dimension of a linear space $\Lambda$ in $\mathbb P^{mn-1}$ whose intersection with the Segre variety $\Sigma_{m-1,n-1}$ does not have any real points.

As the answer by Sasha to the linked question on M.SE explains, for any $\Lambda$ of codimension $\operatorname{dim}\left(\Sigma_{m-1,n-1}\right)=m+n-2$, the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ has nonempty set of complex points.

Now suppose there is a $\Lambda$ of codimension $\leq m+n-2$ such that $\Lambda \cap \Sigma_{m-1,n-1}$ has no real points. Then clearly there is such a $\Lambda$ of codimension equal to $m+n-2$. I claim* that by perturbing $\Lambda$, we can arrange that the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ still has no real points, and each complex point of intersection has multiplicity 1.

Under these hypotheses, the number of complex intersection points equals the degree of $\Sigma_{m-1,n-1}$, which equals ${m+n-2 \choose m-1}$. Now complex conjugation acts on this set of intersection points with no fixed point (because a fixed point would be real). So the degree of the Segre variety must be even.

Contrapositively, if the number ${m+n-2 \choose m-1}$ is odd, then the maximal dimension of a disjoint real subspace is the same as the maximal dimension of a disjoint complex subspace, i.e. $(m-1)(n-1)$.

Unfortunately, the set of pairs $(m,n)$ satisfying this condition seems to have has asymptotic density zero.

*: The set of linear subspaces intersecting $\Sigma$ transversely is a dense open set of the real Grassmannian, while the set of linear subspaces whose intersection has no real points is a Euclidean open set. So their intersection should be nonempty.

Here's a fairly obvious comment about an easy case of this problem. I post it as answer because it's too long for the comment box.

In projective terms, you are looking for the largest dimension of a linear space $\Lambda$ in $\mathbb P^{mn-1}$ whose intersection with the Segre variety $\Sigma_{m-1,n-1}$ does not have any real points.

As the answer by Sasha to the linked question on M.SE explains, for any $\Lambda$ of codimension $\operatorname{dim}\left(\Sigma_{m-1,n-1}\right)=m+n-2$, the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ has nonempty set of complex points.

Now suppose there is a $\Lambda$ of codimension $\leq m+n-2$ such that $\Lambda \cap \Sigma_{m-1,n-1}$ has no real points. Then clearly there is such a $\Lambda$ of codimension equal to $m+n-2$. I claim* that by perturbing $\Lambda$, we can arrange that the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ still has no real points, and each complex point of intersection has multiplicity 1.

Under these hypotheses, the number of complex intersection points equals the degree of $\Sigma_{m-1,n-1}$, which equals ${m+n-2 \choose m-1}$. Now complex conjugation acts on this set of intersection points with no fixed point (because a fixed point would be real). So the degree of the Segre variety must be even.

Contrapositively, if the number ${m+n-2 \choose m-1}$ is odd, then the maximal dimension of a disjoint real subspace is the same as the maximal dimension of a disjoint complex subspace, i.e. $(m-1)(n-1)$.

*: The set of linear subspaces intersecting $\Sigma$ transversely is a dense open set of the real Grassmannian, while the set of linear subspaces whose intersection has no real points is a Euclidean open set. So their intersection should be nonempty.

Here's a fairly obvious comment about an easy case of this problem. I post it as answer because it's too long for the comment box.

In projective terms, you are looking for the largest dimension of a linear space $\Lambda$ in $\mathbb P^{mn-1}$ whose intersection with the Segre variety $\Sigma_{m-1,n-1}$ does not have any real points.

As the answer by Sasha to the linked question on M.SE explains, for any $\Lambda$ of codimension $\operatorname{dim}\left(\Sigma_{m-1,n-1}\right)=m+n-2$, the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ has nonempty set of complex points.

Now suppose there is a $\Lambda$ of codimension $\leq m+n-2$ such that $\Lambda \cap \Sigma_{m-1,n-1}$ has no real points. Then clearly there is such a $\Lambda$ of codimension equal to $m+n-2$. I claim* that by perturbing $\Lambda$, we can arrange that the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ still has no real points, and each complex point of intersection has multiplicity 1.

Under these hypotheses, the number of complex intersection points equals the degree of $\Sigma_{m-1,n-1}$, which equals ${m+n-2 \choose m-1}$. Now complex conjugation acts on this set of intersection points with no fixed point (because a fixed point would be real). So the degree of the Segre variety must be even.

Contrapositively, if the number ${m+n-2 \choose m-1}$ is odd, then the maximal dimension of a disjoint real subspace is the same as the maximal dimension of a disjoint complex subspace, i.e. $(m-1)(n-1)$.

Unfortunately, the set of pairs $(m,n)$ satisfying this condition seems to have asymptotic density zero.

*: The set of linear subspaces intersecting $\Sigma$ transversely is a dense open set of the real Grassmannian, while the set of linear subspaces whose intersection has no real points is a Euclidean open set. So their intersection should be nonempty.

Here's a fairly obvious comment about an easy case of this problem. I post it as answer because it's too long for the comment box.

In projective terms, you are looking for the largest dimension of a linear space $\Lambda$ in $\mathbb P^{mn-1}$ whose intersection with the Segre variety $\Sigma_{m-1,n-1}$ does not have any real points.

As the answer by Sasha to the linked question on M.SE explains, for any $\Lambda$ of codimension $\operatorname{dim}\left(\Sigma_{m-1,n-1}\right)=m+n-2$, the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ has nonempty set of complex points.

Now suppose there is a $\Lambda$ of codimension $\leq m+n-2$ such that $\Lambda \cap \Sigma_{m-1,n-1}$ has no real points. Then clearly there is such a $\Lambda$ of codimension equal to $m+n-2$. I claim* that by perturbing $\Lambda$, we can arrange that the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ still has no real points, and each complex point of intersection has multiplicity 1.

Under these hypotheses, the number of complex intersection points equals the degree of $\Sigma_{m-1,n-1}$, which equals ${m+n-2 \choose m-1}$. Now complex conjugation acts on this set of intersection points with no fixed point (because a fixed point would be real). So the number of complex intersection points must be even.

Contrapositively, if the number ${m+n-2 \choose m-1}$ is odd, then the maximal dimension of a disjoint real subspace is the same as the maximal dimension of a disjoint complex subspace, i.e. $(m-1)(n-1)$.

*: The set of linear subspaces intersecting $\Sigma$ transversely is a dense open set of the real Grassmannian, while the set of linear subspaces whose intersection has no real points is a Euclidean open set. So their intersection should be nonempty.

Here's a fairly obvious comment about an easy case of this problem. I post it as answer because it's too long for the comment box.

In projective terms, you are looking for the largest dimension of a linear space $\Lambda$ in $\mathbb P^{mn-1}$ whose intersection with the Segre variety $\Sigma_{m-1,n-1}$ does not have any real points.

As the answer by Sasha to the linked question on M.SE explains, for any $\Lambda$ of codimension $\operatorname{dim}\left(\Sigma_{m-1,n-1}\right)=m+n-2$, the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ has nonempty set of complex points.

Now suppose there is a $\Lambda$ of codimension $\leq m+n-2$ such that $\Lambda \cap \Sigma_{m-1,n-1}$ has no real points. Then clearly there is such a $\Lambda$ of codimension equal to $m+n-2$. I claim* that by perturbing $\Lambda$, we can arrange that the intersection $\Lambda \cap \Sigma_{m-1,n-1}$ still has no real points, and each complex point of intersection has multiplicity 1.

Under these hypotheses, the number of complex intersection points equals the degree of $\Sigma_{m-1,n-1}$, which equals ${m+n-2 \choose m-1}$. Now complex conjugation acts on this set of intersection points with no fixed point (because a fixed point would be real). So the degree of the Segre variety must be even.

Contrapositively, if the number ${m+n-2 \choose m-1}$ is odd, then the maximal dimension of a disjoint real subspace is the same as the maximal dimension of a disjoint complex subspace, i.e. $(m-1)(n-1)$.

*: The set of linear subspaces intersecting $\Sigma$ transversely is a dense open set of the real Grassmannian, while the set of linear subspaces whose intersection has no real points is a Euclidean open set. So their intersection should be nonempty.