Problem Analysis: We need to determine the relationship between two expressions, $x = n^{\log_{10}(m)}$ and $y = m^{\log_{10}(n)}$, given that $m$ and $n$ are distinct positive real numbers with $m > n$. This requires understanding properties of logarithms and exponents.

Comparing x and y using Logarithm Properties

1. Analyze expression x: Let $x = n^{\log_{10}(m)}$. To simplify, we take the base-10 logarithm of both sides:

   $ \log_{10}(x) = \log_{10}(n^{\log_{10}(m)}) $

   Using the logarithm power rule, $\log(a^b) = b \log(a)$, we get:

   $ \log_{10}(x) = (\log_{10}(m)) \cdot (\log_{10}(n)) $

2. Analyze expression y: Let $y = m^{\log_{10}(n)}$. Similarly, take the base-10 logarithm of both sides:

   $ \log_{10}(y) = \log_{10}(m^{\log_{10}(n)}) $

   Applying the logarithm power rule:

   $ \log_{10}(y) = (\log_{10}(n)) \cdot (\log_{10}(m)) $

3. Compare logarithms: By comparing the results from Step 1 and Step 2:

   $ \log_{10}(x) = (\log_{10}(m)) \cdot (\log_{10}(n)) $

   $ \log_{10}(y) = (\log_{10}(n)) \cdot (\log_{10}(m)) $

   Since the order of multiplication does not matter (i.e., $a \cdot b = b \cdot a$), we have:

   $ \log_{10}(x) = \log_{10}(y) $

4. Conclude the relation: The logarithm function is a one-to-one function. This means if $\log_{10}(x) = \log_{10}(y)$, then $x$ must be equal to $y$. The conditions that $m, n$ are positive distinct real numbers ensure that these expressions are well-defined.

   Therefore, the relation is $x = y$.