How does tip-tilt mirrors correct such issue? What is the physical origin of the error?
Isoplanatism commonly refers to a region of angles over which a ground-based telescope observes effectively the same atmospheric turbulence (e.g. an "isoplanatic patch"), such that a laser guide star provides effective correction of atmospheric seeing.
an-isoplanatism refers to a lack of isoplanatism, or a way in which the science target and the reference laser guide star differ.
A single laser guide star does not provide knowledge of tip or tilt errors since the up-going beam experiences tilts in addition the down-going signal; these two sets of tilts are entangled and in the case where the transmitter and receiver are identical the laser guide star appears stationary. See a detailed overview and history of this problem the introduction of Ragazzoni 1996, http://adsabs.harvard.edu/full/1996ApJ...465L..73R. This is commonly solved by observing a "natural guide star" position, since its light only passes through the atmosphere once, and using that signal to maintain the position of the target star by changing the angle a small ("tip-tilt") mirror.