Studying certain semidefinite programs arising in spectral graph theory, I discovered semi-definite primal/dual pair satisfying strong duality but not strict complementarity.

Let $E=\{12,23,34\}$ (the edge set of the path graph $P_3$) and $E^{ij}:=(\mathbf e_i-\mathbf e_j)^\top(\mathbf e_i-\mathbf e_j)$. We use the inner product $
\def\<{\langle}\def\>{\rangle}\<X,Y\>:=\mathrm{tr}(XY)$ on the space of symmetric matrices $\mathbf S^n$.

$$
\llap{\mathrm{(P)}\qquad}
\boxed{\begin{array}{rlr}
p^*=\max & \sum_{ij\in E} w_{ij} \\
\mathrm{s.t.} & I-\sum_{ij\in E} w_{ij}E^{ij} \succeq0 \\
              & w_{ij}\ge 0,\quad ij\in E
\end{array}}$$

$$
\llap{\mathrm{(D)}\qquad}
\boxed{\begin{array}{rlr}
d^*=\min & \mathrm{tr}(X) \\
\mathrm{s.t.} & \<X,E^{ij}\>\ge 1,\quad ij\in E \\
              & X\succeq 0
\end{array}}$$

The common optimal value is $p^*=d^*=1$, obtained for

$$w_{12}=w_{34}=\frac12,\quad w_{23}=0,$$

$$X=\frac14\begin{pmatrix}
\phantom+1 & -1 & \phantom+1 & -1 \\
-1 & \phantom+1 & -1 & \phantom+1 \\
\phantom+1 & -1 & \phantom+1 & -1 \\
-1 & \phantom+1 & -1 & \phantom+1
\end{pmatrix} 
= \frac14 (\phantom+1,-1,\phantom+1,-1)^\top(\phantom+1,-1,\phantom+1,-1).$$

Both are the unique solutions of their respective problem (this can easily be seen when interpreting $\mathrm{(D)}$ as an embedding problem, but I will leave this out here). As seen, the dual positive semi-definite matrix $X$ has rank $1$. However, the corresponding matrix

$$Z:=I-\sum_{ij\in E} w_{ij}E^{ij} = 
\frac12 \begin{pmatrix} 
1 & 1 &   &   \\
1 & 1 &   &   \\ 
  &   & 1 & 1 \\
  &   & 1 & 1
\end{pmatrix}$$

is of rank $2$. Hence $\rank(X)+\rank(Z)=3<4$ and strict complementarity is not satisfied.