You might be interested in controlled version of $-I$. Despite the fact that you can neglect global phase in case of non-controlled gates, you cannot do so in case of controlled version.

The controled gate $-I$ is described by matrix
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1\\
\end{pmatrix}.

This gate set a phase to $\pi$ (note that $\mathrm{e}^{i\pi} = -1$) if control qubit is in state $|1\rangle$.

To implement the gate simply put $Z$ gate on first qubit (i.e. control qubit) and nothing (i.e. identity operator) on second qubit (i.e. target qubit). You can check that the matrix above is really equal to $Z \otimes I$ and hence the proposed construction really implements the requested gate.

2

The answer to this question (https://quantumcomputing.stackexchange.com/questions/2477/phase-shift-gate-in-qiskit) also gives a construction that does not need ancilla. It's construction is $\hat P_h(\theta)=\hat U_1(\theta)\hat X\hat U_1(\theta)\hat X$, and in your case, replace $\theta$ with $\frac{\pi}{2}$

– Yitian Wang – 2020-11-24T11:43:15.207They seem to be using "inverse" in the sense of additive inverse, but there really isn't much meaning to the additive inverse in QM. "Negation gate" would probably be a better phrasing. – Tyberius – 2020-11-24T20:18:41.383