The T238/T238+ support an external Real Time Clock. It is a one-wire device called the DS1994L. The device is an iButton, a metal can looking roughly like a large lithium battery. It can be purchased from any decent distributor carrying 1-wire devices. A source (not necessarily the cheapest) is from Maxim directly and can be found on the links page of this website.
When you order the DS1994L, you will need a reader for it as well--both for your PC as well as the weather station. The most convenient reader is the DS1402D-DR8. That will plug directly into the same RJ11 connector used by the T238s. For connectivity to the PC, you will need the DS9097U-S09 serial to RJ-11 adapter. If you don't have a serial port, the DS9490R appears to be the USB adapter, but I can't personally vouch for it. The PC software needed to set the time on the DS1994 is on Maxim's website called OneWireViewer.
The general process is that you use your PC to set the time on the DS1994 and then the T238s read that time on power-up. There is no software on the T238s to set the time on the DS1994. There are two very important things to keep in mind when setting the time on the DS1994. First, set the to UTC. You need both UTC and local time for weather operations. The T238s assume the DS1994 is UTC and then you set the offset to local time in the T238 menus. Second, and this is really, really important, whatever you do, do NOT set the write protect bits for the time registers in the DS1994. This is a security measure so if used in an un-trusted environment, bad guys cannot alter the time in the device. It also means that if you set them, you will NEVER be able to re-set the time ever.
If you don't already have one, a simple 1-to-many RJ11 adapter intended to plug many phone devices into one outlet is a nice way to expand your one-wire bus. Plug it into your T238 then the reader and your other weather devices into the other ports. On power-up, the T238 will go search for a DS1994. If it finds one, it will read the time, use it for UTC, apply your programmed offset and use that time for local time.
It is a bit of a pain to set up, but once done, it is a set-and-forget operation.