Due to the (co-)linearity of the bracket $\langle-,-\rangle$, it suffices to consider
$$
\left\langle\int^-_0\Theta_sds,\int^-_0\tilde{\Gamma}_sdW_s\right\rangle_t+\left\langle\int^-_0\Gamma_sds,\int^-_0\tilde{\Gamma}_sdW_s\right\rangle_t.
$$
Thus, we shall prove
$$
\left\langle\int^-_0\Theta_sds,\int^-_0\tilde{\Gamma}_sdW_s\right\rangle_t=0,\tag{1}
$$
and
$$
\left\langle\int^-_0\Gamma_sds,\int^-_0\tilde{\Gamma}_sdW_s\right\rangle_t=\int^t_0\Gamma_s\tilde{\Gamma}_sds.\tag{2}
$$
The bracket (quadratic covariation) process is defined by
$$
\langle X,Y\rangle_t:=X_tY_t-\int^t_0X_sdY_s-\int^t_0Y_sdX_s,
$$
where the co-linearity is evident. This process is characterized as the limit in probability:
$$
X_0Y_0+\sum_{j=0}^{n-1}(X_{t_{j+1}}-X_{t_j})(Y_{t_{j+1}}-Y_{t_j})\xrightarrow{\operatorname{P}}\langle X,Y\rangle_t,
$$
where $0=t_0<t_1<\cdots<t_n=t$ is a subdivision of the interval $[0,t]$, and the limit is taken as $n\to\infty$ and $\displaystyle|\pi_n|:=\max_{j=0,\cdots,n-1}|t_{j+1}-t_j|\to0$. Therefore, our goal is to identify $\langle X,Y\rangle$ via the above limit.
Proving Equality (1)
Let $A_t:=\int^t_0\Theta_sds$ and $\tilde{M}_t:=\int^t_0\tilde{\Gamma}_sds$. Both have continuous paths, thus in particular, they have uniformly continuous paths on each compact interval $[0,t]$. Therefore, for any subdivision $\pi_n:=\{t_j\}_{j=0}^n\subset[0,t]$, we obtain
\begin{align*}
&\sum_{j=0}^{n-1}(A_{t_{j+1}}-A_{t_j})(M_{t_{j+1}}-M_{t_j})\\
&\quad\le\sup_{\substack{|s-u|\le|\pi_n|\\s,u\in[0,t]}}|M_u-M_s|\sum_{j=0}^{n-1}|A_{t_{j+1}}-A_{t_j}|\\
&\quad\le\left(\sup_{\substack{|s-u|\le|\pi_n|\\s,u\in[0,t]}}|M_u-M_s|\right)\left(\sum_{j=0}^{n-1}|A_{t_{j+1}}-A_{t_j}|\right).
\end{align*}
Since the paths of $A_t$ is of bounded variation, the right-hand side converges to $0$ as $n\to\infty$ with $|\pi_n|\to0$, thus concluding $\langle A,\tilde{M}\rangle_t=0$.
Proving Equality (2)
To identify the limit of
$$
\sum_{j=0}^n\left(\int^{t_{j+1}}_{t_j}\Gamma_sdW_s\right)\left(\int^{t_{j+1}}_{t_j}\tilde{\Gamma}_sdW_s\right),
$$
we have to use the definition of the Itô integral for simple processes $\Gamma_s,\tilde{\Gamma}_s$, and then extend the result to the general settings using approximation arguments. Famously, the limit is $\int^t_0\Gamma_s\tilde{\Gamma}_sds$.
