  Exact-exchange energy density and energy density of a semilocal density
functional approximation are two key ingredients for modeling the static
correlation, a strongly nonlocal functional of the density, through a local
hybrid functional. Because energy densities are not uniquely defined, the
conventional (Slater) exact-exchange energy density
$e_\mathrm{x}^\mathrm{ex(conv)}$ is not necessarily well-suited for local
mixing with a given semilocal approximation. We show how to transform
$e_\mathrm{x}^\mathrm{ex(conv)}$ in order to make it compatible with an
arbitrary semilocal density functional, taking the nonempirical
meta-generalized gradient approximation of Tao, Perdew, Staroverov, and
Scuseria (TPSS) as an example. Our additive gauge transformation function
integrates to zero, satisfies exact constraints, and is most important where
the density is dominated by a single orbital shape. We show that, as expected,
the difference between semilocal and exact-exchange energy densities becomes
more negative under bond stretching in He$_2^{+}$ and related systems. Our
construction of $e_\mathrm{x}^\mathrm{ex(conv)}$ by a
resolution-of-the-identity method requires uncontracted basis functions.
