  We investigate half-BPS Type IIB supergravity solutions with spacetime
geometry $AdS_2\times S^6$ warped over a Riemann surface $\Sigma$. The general
local solution was obtained in earlier work in terms of two holomorphic
functions $\mathcal A_\pm$ on $\Sigma$. In the first part of this paper we seek
global solutions corresponding to the near-horizon behavior of $(p,q)$-string
junctions. We identify the type of singularity in $\mathcal A_\pm$ needed at
the boundary of $\Sigma$ to match the solutions locally onto the classic
$(p,q)$-string solution. We construct solutions with multiple $(p,q)$-strings,
but the existence of geodesically complete solutions remains unsettled. In a
second part we construct multi-parameter families of non-compact globally
regular and geodesically complete solutions with asymptotic regions and an
$AdS_2$ throat which caps off smoothly.
