Let's say we have $l_1$ norm minimization $||Ax-b||_1$ with matrix $A$ with consecutive ones property. Now, we can construct a mathematical formulation to solve this problem:
\begin{align} \text{min} \, &\sum\limits_{i=1}^m s_i\\ &(1) \sum\limits_{j=1}^n a_{ij} x_j + s_i \geq b_i,\quad \forall i,\\ &(2) \sum\limits_{j=1}^n a_{ij} x_j - s_i \leq b_i,\quad \forall i\\ \end{align}
Now its clear that $[A,I]$ (matrix of constraint 1) and (matrix of constraint 2) $[A,-I]$ (proof can be done with using Laplacean expansion) are TU matrices, but its not clear that $\begin{bmatrix} A & I \\ A & -I \end{bmatrix}$ is which is the constraint matrix of this problem. I have seen empirically that I only get integer values for $x$ for all instances of $b$ I have tried, but it not clear to me that the constraint matrix is TU, is it?
No it is not, unless $A$ has some other property. As a counter-example, you can take $A=(1)$ and $I$ of size $1$, you get determinant $-2$.
About your MIP...
If $Ax=b$ is feasible, then you get as obvious minimum $s=0$, and the constraint matrix becomes $(A^*,A^*)^*$ ($^*$ denoting transposition) which is TU (it verifies the consecutive ones property).
Do you still get only integer values when ensuring $Ax=b$ not feasible?