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 $\mathscr L_g=\sqrt{-g}R$.  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?

(Aside: I understand that this might require a pseudo tensor equation that satisfies $\partial_\mu t^{\mu\nu}$ globally, but in any such globally conserved quantity I've seen derived, using for example Noerther's second theorem, the field terms go away leaving a pseudo tensor expressed only in (derivatives of) the metric $g^{\mu\nu}$.  

But surely there must be some expression, however complicated, for the momentum as it winds off a straight path in curved space, since all we are doing is replacing a straight line by a geodesic, and the total momentum ought to remain constant, or change according to a shift in potential energy which balances the equation.)