Let us begin with a Lagrangian of the form

$$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$

where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$ Suppose as well that there are no Killing vectors associated to the metric $g^{\mu\nu}$ except for, say, a global timelike Killing vector if it helps the argument.

Associated to $\mathscr L$ is a locally conserved set of 10 currents from the Poincare group:

$$T^{\mu\nu}=\partial^\mu\phi(x)\partial^\nu\phi(x)-g^{\mu\nu}\mathscr L$$

for each spacetime translation, and $\epsilon_{\alpha\beta}x^\alpha T^{\mu\beta}$ for each spacetime rotation.

Locally we have $$\nabla_\mu T^{\mu\nu}=0$$ so these quantities are conserved only locally.

My question is, what is the obstacle to patching these locally conserved quantities together to make a globally conserved quantity:

$$Q=\int T^{0 \nu}f_\nu \;d^3x$$

with

$$dQ/dt=0$$

where $f_\nu$ might be a gluing function connecting the momentum flowing out of one patch of infinitesimal volume and into another?

(Edit: I realize there may not be a tensor associated to this conserved quantity but even a pseudo tensor involving only the fields would be satisfying, if it exists. So for example, to get the ball rolling, we can start with an object of the form

$M^{\lambda\mu\nu}=\frac 12\int_{a^\mu}^{x^\lambda}ds T^{\mu\nu}(s)-\frac 12\int_{a^\mu}^{x^\mu}ds T^{\lambda\nu}(s)$,

and then set $t^{\mu\nu}=\partial_\lambda M^{\lambda\mu\nu}$.

$t^{\mu\nu}$ is a psuedo tensor that is conserved $\partial_\mu t^{\mu\nu}=0$ generically by the antisymmetry in $\lambda,\ \mu$, and as well

$t^{\mu\nu}=T^{\mu\nu}(x)+\frac 12\int^{x^\mu}ds\ \partial_\lambda T^{\lambda\nu}(s)+\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu})$

In flat space this quantity is almost conserved up to the boundary term $\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu})$ where $\delta^\mu_\lambda$ is a Kronecker delta.

Of course, the boundary term ruins it. That, and the lack of symmetry in mu and nu, but this should give the idea of what could with a better choice of starting point.)