My wife and I are on the same Roam Like Home plan, currently visiting Mexico. When she texts me, I receive the message instantaneously. When I reply, she doesn't receive the message instantaneously. It seems she has to be in a place that's WiFi connected. What could be the reason for this. We both have iPhone 5s.
Thank you for your post. There are many variables which may be causing the texts to be delivered only when the other phone is connected to WiFi. Is the Data Roaming enabled on the phone which is only receiving the text when connected to WiFi? What is the text bubble's colour? Green is usually for text message and blue is for iMessages.
From the problem you described, it sounds like you are sending an iMessage, which would show up as blue in your texting log, as RogersMoin mentioned. iMessage requires a data connection to send or receive messages which explains why you would only be receiving the messages once you are near a WiFi location.
You can do two things to solve your problem. The first would be to turn data roaming on, which would connect you to cellular data and allow you to receive the iMessage as soon as it comes in. Your other option would be to open the Settings application by tapping the "Settings" icon on your iPhone's home screen. Tap the "Messages" row to open the Messages screen. Tap the switch next to "iMessage" so that it reads "Off." Your iPhone will now send all messages in text message format instead of using the iMessage service, so you do not require a data connection.