As the others' answers and comments have pointed out, the most common written form of this phrase is 真的, even when it is used as an adverb. 真地 is rarely seen, and may even be considered "weird" or "incorrect".
Wiktionary lists 真的 (with this particular written form) as both an adverb and an adjective (真的 - Wiktionary), and doesn't have a page for 真地.
In 《现代汉语八百词》(Eight Hundred Words in Modern Chinese), editor-in-chief 吕叔湘(Lü Shuxiang), the entry of 真 is as follows:
- c) 修饰动词、形容词以带‘的’为常。（我真的要走，不骗你 | 他真的不想去 | 你真的有意见？ | 事情真的很顺利 | 昨天来的真是老李吗？）
- d) ‘真的’修饰全句，可用在主语前，后面有停顿。（真的，手术后恢复的比别人都快 | 真的，我们厂今年提前两个月完成了全年计划）
- c) Usually with "的" when modifying verbs or adjectives. (I'm really going to go, no kidding | He really doesn't want to go | Do you seriously have an opinion on this? | things went really well | Was it really Li who came yesterday?)
- d) "真的" can modify the whole sentence. It can be used before the subjective with a following pause. (Really, [dropped subjective] recovered faster than all the other people after the operation | I'm serious! Our factory accomplished our annual goal two weeks before the year ends this year)
The use of 真的 but not 真地 as an adverb is (almost) uncontroversial, but also, on the surface, a blatant outlier of the usual 得-的-地 distinction rules taught in schools. So try and consider this a convention that doesn't conform to these rules. musialmi's comment is right: it can be considered as a two-character adverb that happens to contain the character 的. 真地, on the other hand, is not used, or even if it used, it was probably because of hyper-correction influenced by 得-的-地.
[I'll digress a little in this paragraph and talk about another issue that musialmi brought forth in their comment. Probably not all people will agree with musialmi on 十分的. Words like 十分, 非常, 相当 are adverbs of degree; it is yet universally agreed upon whether it should be 的 or 地 when you want to add a de modifier after them. Both 非常地 and 非常的, for example, can be seen in literary works. In this case, if you don't want to be nitpicked by grammar police, sticking with 地 would be a safer approach, because "地 for adverbials" is agreed by mostly everyone (again, there are exceptions like 真的太可爱了). But more natural will be not putting a de modifier after an adverb of degree at all (十分地/的有趣 → 十分有趣), because words like 十分 or 相当 don't need one.]